When you improve a slice's rigidity by folding it, you're experiencing Gaussian curvature.
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This is a blur with a more defined, circular shape than gaussian blur.
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Royen found that he could generalize the GCI to apply not just to Gaussian distributions of random variables but to more general statistical spreads related to the squares of Gaussian distributions, called gamma distributions, which are used in certain statistical tests.
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His WIRED cover story on the Gaussian copula function was later turned into a tattoo.
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In 2017, Cole used algorithms called Gaussian process regressions (GPRs) to generate a brain age for each participant.
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Gaussian curvature reflects the combination of two distinct curvatures (such as on an x-axis and a y-axis).
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Darts thrown at the target will land in a bell curve or "Gaussian distribution" of positions around the center point.
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There are limits to how well an optical phenomenon can be replicated if you are taking shortcuts like Gaussian blurring.
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Update: On additional inquiry, Apple clarified for me that it is in fact not gaussian effect but instead a custom disc blur.
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The use of spatial audio anchored in what seem to be Gaussian spheres that attach sound and (incredible) music to environments, with nested encounter scores inside.
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According to Trick Photoshop, you can find the most dominant color in Photoshop by choosing Filter, Blur then Gaussian Blur and ramping the blur level right up.
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First duplicate your current layer (Layer then Duplicate Layer) then go to Filter, Blur, Gaussian Blur and adjust the level so the details are just getting lost.
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Once it has these nine slices, it can then pick and choose which layers are sharp and which get a gaussian blur blur effect applied to them.
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Researching cotton prices, mathematician Benoit Mandelbrot discovered there were far more very large daily price moves than would be expected if changes followed a normal or Gaussian distribution.
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Is the whole of my being—my Gaussian memories, my anxiety disorders and tics, my capacity for intimacy or destruction—really self-contained within this electrified slushball of meat?
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To illustrate, if I wanted to come up with a random number that fits a normal or Gaussian distribution, I can just refer to the P5 API, which conveniently has a function called randomGaussian().
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It did a great job masking the dirty dishes in the sink behind me, even if the effect appeared to have been designed by a film student who just discovered the Gaussian blur filter.
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If you plot people's weights and heights on an x–y plot, the weights will form a Gaussian bell-curve distribution along the x-axis, and heights will form a bell curve along the y-axis.
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It maintains color nicely, making sure that the quality of light isn't obscured like it is with so many other portrait applications in other phones that just pick a spot and create a circle of standard gaussian or disc blur.
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When the first wave of blurring products hit the market a few years back, primers took precedent, offering up a number of ways in which one could turn their skin into a flattering Gaussian blur of itself before even putting on makeup.
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Teasing out the cosmological triangles and other shapes—which have been named "non-Gaussianities" to contrast them with the Gaussian bell curve of randomly distributed pairs of structures—will require more precise observations of the cosmos than have been made to date.
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Dunn, generalizing an inequality posed three years earlier, conjectured the following: The probability that both Gaussian random variables will simultaneously fall inside the rectangular region is always greater than or equal to the product of the individual probabilities of each variable falling in its own specified range.
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The Gaussian correlation inequality says that the probability that a dart will land inside both the rectangle and the circle is always as high as or higher than the individual probability of its landing inside the rectangle multiplied by the individual probability of its landing in the circle.
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You have people that have a mesh of funk and house like Moon B, and then you have some of the UK boogie like Index and then the Gaussian Curve stuff… It's more ambient stuff, but there's also Uncle Jamm's Army with "Dial-A-Freak," and that's electrofunk.
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Original story reprinted with permission from Quanta Magazine, an editorially independent division of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 295s, was posed in its most elegant form in 20 and has held mathematicians in its thrall ever since.
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Gaussian 70, Gaussian 76, Gaussian 80, Gaussian 82, Gaussian 86, Gaussian 88, Gaussian 90, Gaussian 92, Gaussian 92/DFT, Gaussian 94, Gaussian 98, Gaussian 03, Gaussian 09 and Gaussian 16. Other programs named 'Gaussian XX' were placed among the holdings of the Quantum Chemistry Program Exchange. These were unofficial, unverified ports of the program to other computer platforms.
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Gaussian ARTJames R. Williamson. (1996), Gaussian ARTMAP: A Neural Network for Fast Incremental Learning of Noisy Multidimensional Maps, Neural Networks, 9(5):881-897 and Gaussian ARTMAP use Gaussian activation functions and computations based on probability theory. Therefore, they have some similarity with Gaussian mixture models.
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Spatial stochastic process can become computationally effective and scalable Gaussian process models, such as Gaussian Predictive Processes and Nearest Neighbor Gaussian Processes (NNGP).
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The field of Gaussian rationals is the field of fractions of the ring of Gaussian integers. It consists of the complex numbers whose real and imaginary part are both rational. The ring of Gaussian integers is the integral closure of the integers in the Gaussian rationals. This implies that Gaussian integers are quadratic integers and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation :x^2 +cx+d=0, with and integers.
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Since sums of independent Gaussian random variables are themselves Gaussian random variables, this conveniently simplifies analysis, if one assumes that such error sources are also Gaussian and independent.
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A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes. Note that there are rational primes which are not Gaussian primes.
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A Gaussian fixed point is a fixed point of the renormalization group flow which is noninteracting in the sense that it is described by a free field theory. The word Gaussian comes from the fact that the probability distribution is Gaussian at the Gaussian fixed point. This means that Gaussian fixed points are exactly solvable (trivially solvable in fact). Slight deviations from the Gaussian fixed point can be described by perturbation theory.
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A Gaussian pulse is shaped as a Gaussian function and is produced by a Gaussian filter. It has the properties of maximum steepness of transition with no overshoot and minimum group delay.
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When a parameterised kernel is used, optimisation software is typically used to fit a Gaussian process model. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution.
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The first systematic model chemistry of this type with broad applicability was called Gaussian-1 (G1) introduced by John Pople. This was quickly replaced by the Gaussian-2 (G2) which has been used extensively. The Gaussian-3 (G3) was introduced later.
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A number of fields such as stellar photometry, Gaussian beam characterization, and emission/absorption line spectroscopy work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. There are three unknown parameters for a 1D Gaussian function (a, b, c) and five for a 2D Gaussian function (A; x_0,y_0; \sigma_X,\sigma_Y). The most common method for estimating the Gaussian parameters is to take the logarithm of the data and fit a parabola to the resulting data set.Hongwei Guo, "A simple algorithm for fitting a Gaussian function," IEEE Sign. Proc. Mag.
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The majority of optical tweezers make use of conventional TEM00 Gaussian beams. However a number of other beam types have been used to trap particles, including high order laser beams i.e. Hermite-Gaussian beams (TEMxy), Laguerre-Gaussian (LG) beams (TEMpl) and Bessel beams. Optical tweezers based on Laguerre-Gaussian beams have the unique capability of trapping particles that are optically reflective and absorptive.
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In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
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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.
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Independent component analysis (ICA) is a technique for forming a data representation using a weighted sum of independent non-Gaussian components. The assumption of non-Gaussian is imposed since the weights cannot be uniquely determined when all the components follow Gaussian distribution.
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22, no. 9, pp. 1435-1445, Sep. 2011. by using Gaussian priors, whereby a Gaussian process model with ESN-driven kernel function is obtained.
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The discrete Gaussian kernel (solid), compared with the sampled Gaussian kernel (dashed) for scales t=0.5,1,2,4. One may ask for a discrete analog to the Gaussian; this is necessary in discrete applications, particularly digital signal processing. A simple answer is to sample the continuous Gaussian, yielding the sampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the properties of the continuous function, and can lead to undesired effects, as described in the article scale space implementation.
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In a Gaussian pyramid, subsequent images are weighted down using a Gaussian average (Gaussian blur) and scaled down. Each pixel containing a local average corresponds to a neighborhood pixel on a lower level of the pyramid. This technique is used especially in texture synthesis.
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The cross-over trajectory tangent to the green arrows connects the non-Gaussian to the Gaussian fixed point and plays the role of a separatrix.
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Filtering involves convolution. 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 Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.
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Note that although this model is termed a "Gaussian chain", the distribution function is not a gaussian (normal) distribution. The end-to-end distance probability distribution function of a Gaussian chain is non-zero only for r > 0\. In fact, the Gaussian chain's distribution function is also unphysical for real chains, because it has a non-zero probability for lengths that are larger than the extended chain.
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The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
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In particular, by choosing a negative, it is evident that the Weierstrass transform of a Gaussian function is again a Gaussian function, but a "wider" one.
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EnKFs rely on the Gaussian assumption, although they in practice are used for nonlinear problems, where the Gaussian assumption may not be satisfied. Related filters attempting to relax the Gaussian assumption in EnKF while preserving its advantages include filters that fit the state pdf with multiple Gaussian kernels, filters that approximate the state pdf by Gaussian mixtures, a variant of the particle filter with computation of particle weights by density estimation, and a variant of the particle filter with thick tailed data pdf to alleviate particle filter degeneracy.
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Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. The final nonzero remainder is , the Gaussian integer of largest norm that divides both and ; it is unique up to multiplication by a unit, or . Many of the other applications of the Euclidean algorithm carry over to Gaussian integers. For example, it can be used to solve linear Diophantine equations and Chinese remainder problems for Gaussian integers; continued fractions of Gaussian integers can also be defined.
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A Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process. An important special case of a GRF is the Gaussian free field. With regard to applications of GRFs, the initial conditions of physical cosmology generated by quantum mechanical fluctuations during cosmic inflation are thought to be a GRF with a nearly scale invariant spectrum.
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While no amount of delay can make a theoretical Gaussian filter causal (because the Gaussian function is non-zero everywhere), the Gaussian function converges to zero so rapidly that a causal approximation can achieve any required tolerance with a modest delay, even to the accuracy of floating point representation.
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Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression, i.e., for fitting a curve through noisy data.
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His research group developed the quantum chemistry composite methods such as Gaussian-1 (G1) and Gaussian-2 (G2). In 1991, Pople stopped working on Gaussian and several years later he developed (with others) the Q-Chem computational chemistry program.Pople's Q-Chem page Prof. Pople's departure from Gaussian, along with the subsequent banning of many prominent scientists, including himself, from using the software gave rise to considerable controversy among the quantum chemistry community.
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An example of Gaussian Process Regression (prediction) compared with other regression models.The documentation for scikit-learn also has similar examples. A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. Given any set of N points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.
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The solution can be approximated using geometric spanners. In number theory, the unsolved Gaussian moat problem asks whether or not minimax paths in the Gaussian prime numbers have bounded or unbounded minimax length. That is, does there exist a constant such that, for every pair of points and in the infinite Euclidean point set defined by the Gaussian primes, the minimax path in the Gaussian primes between and has minimax edge length at most ?.
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Another photonic implementation of boson sampling concerns Gaussian input states, i.e. states whose quasiprobability Wigner distribution function is a Gaussian one. The hardness of the corresponding sampling task can be linked to that of scattershot boson sampling. Namely, the latter can be embedded into the conventional boson sampling setup with Gaussian inputs.
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For solution of the multi- output prediction problem, Gaussian process regression for vector-valued function was developed. In this method, a 'big' covariance is constructed, which describes the correlations between all the input and output variables taken in N points in the desired domain. This approach was elaborated in detail for the matrix-valued Gaussian processes and generalised to processes with 'heavier tails' like Student-t processes. Inference of continuous values with a Gaussian process prior is known as Gaussian process regression, or kriging; extending Gaussian process regression to multiple target variables is known as cokriging.
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In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns linear systems driven by additive white Gaussian noise. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Output measurements are assumed to be corrupted by Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector.
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A Wiener process (aka Brownian motion) is the integral of a white noise generalized Gaussian process. It is not stationary, but it has stationary increments. The Ornstein–Uhlenbeck process is a stationary Gaussian process. The Brownian bridge is (like the Ornstein–Uhlenbeck process) an example of a Gaussian process whose increments are not independent.
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The norm of a Gaussian integer x + yi is the number x2 + y2. Thus, the Pythagorean primes (and 2) occur as norms of Gaussian integers, while other primes do not. Within the Gaussian integers, the Pythagorean primes are not considered to be prime numbers, because they can be factored as :p = (x + yi)(x − yi).
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In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval (0,\infty)) as a result of censoring.
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In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem.
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Instead, spacing of latitudes is defined by Gaussian quadrature. By contrast, in the "normal" geographic latitude-longitude grid, gridpoints are equally spaced along both latitudes and longitudes. Gaussian grids also have no grid points at the poles. In a regular Gaussian grid, the number of gridpoints along the longitudes is constant, usually double the number along the latitudes.
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Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.
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This Gaussian process is called the Neural Network Gaussian Process (NNGP). It allows predictions from Bayesian neural networks to be more efficiently evaluated, and provides an analytic tool to understand deep learning models.
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This gives an output pulse shaped like a Gaussian function.
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A Gaussian or a Lanczos filter are considered good compromises.
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ImageFileReader() reader.SetFileName(sys.argv[1]) image = reader.Execute() pixelID = image.GetPixelID() gaussian = sitk.
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The black curve is the corresponding intensity. A 5 mW green laser pointer beam profile, showing the TEM00 profile. In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot.
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Gaussian is a general purpose computational chemistry software package initially released in 1970 by John Pople and his research group at Carnegie Mellon University as Gaussian 70.W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Gaussian 70 (Quantum Chemistry Program Exchange, Program No. 237, 1970) It has been continuously updated since then. The name originates from Pople's use of Gaussian orbitals to speed up molecular electronic structure calculations as opposed to using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree–Fock calculations. The current version of the program is Gaussian 16.
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If we regard the ring of Gaussian integers, we get the case and , and can ask (WLOG) for which the number is a Gaussian prime which will then be called a Gaussian Mersenne prime.Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages) is a Gaussian prime for the following : :2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers. As for all Gaussian primes, the norms (that is, squares of absolute values) of these numbers are rational primes: :5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... .
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The resulting linear circuit matrix can be solved with Gaussian elimination.
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A rectified Gaussian distribution is semi-conjugate to the Gaussian likelihood, and it has been recently applied to factor analysis, or particularly, (non- negative) rectified factor analysis. Harva proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng proposed an infinite rectified factor model coupled with its Gibbs sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for reconstruction of gene regulatory networks.
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In Gaussian process regression, also known as Kriging, a Gaussian prior is assumed for the regression curve. The errors are assumed to have a multivariate normal distribution and the regression curve is estimated by its posterior mode. The Gaussian prior may depend on unknown hyperparameters, which are usually estimated via empirical Bayes. The hyperparameters typically specify a prior covariance kernel.
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Intensity of a plane wave diffracted through an aperture with a Gaussian profile The diffraction pattern obtained given by an aperture with a Gaussian profile, for example, a photographic slide whose transmissivity has a Gaussian variation is also a Gaussian function. The form of the function is plotted on the right (above, for a tablet), and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings.Hecht, 2002, Figure 11.33 This technique can be used in a process called apodization—the aperture is covered by a Gaussian filter, giving a diffraction pattern with no secondary rings. The output profile of a single mode laser beam may have a Gaussian intensity profile and the diffraction equation can be used to show that it maintains that profile however far away it propagates from the source.
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Gaussian processes are thus useful as a powerful non- linear multivariate interpolation tool. Gaussian process regression can be further extended to address learning tasks in both supervised (e.g. probabilistic classification) and unsupervised (e.g. manifold learning) learning frameworks.
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In the geostatistics community Gaussian process regression is also known as Kriging.
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For practical regression and prediction needs, Student's t-processes were introduced, that are generalisations of the Student t-distributions for functions. A Student's t-process is constructed from the Student t-distributions like a Gaussian process is constructed from the Gaussian distributions. For a Gaussian process, all sets of values have a multidimensional Gaussian distribution. Analogiusly, X(t) is a Student t-process on an interval I=[a,b] if the correspondent values of the process X(t_1),...,X(t_n) (t_i \in I) have a joint multivariate Student t-distribution.
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Carl Friedrich Gauss Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units.One of many examples of using the term "cgs units" to refer to Gaussian units is: Lecture notes from Stanford University The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of cgs with conflicting definitions of electromagnetic quantities and units.
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Kriging starts with a prior distribution over functions. This prior takes the form of a Gaussian process: N samples from a function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location of two points. A set of values is then observed, each value associated with a spatial location. Now, a new value can be predicted at any new spatial location, by combining the Gaussian prior with a Gaussian likelihood function for each of the observed values.
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In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kallianpur-Sato-Stefan and Dudley-Feldman-le Cam. There is the earlier result due to H. Satô (1969) H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure, 1969. which proves that "any Gaussian measure on a separable Banach space is an abstract Wiener measure in the sense of L. Gross".
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F. Yang, S. Wang, and C. Deng, "Compressive sensing of image reconstruction using multi-wavelet transform", IEEE 2010 The current smallest upper bounds for any large rectangular matrices are for those of Gaussian matrices.B. Bah and J. Tanner "Improved Bounds on Restricted Isometry Constants for Gaussian Matrices" Web forms to evaluate bounds for the Gaussian ensemble are available at the Edinburgh Compressed Sensing RIC page.
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Gaussian elimination can be performed over any field, not just the real numbers. Buchberger's algorithm is a generalization of Gaussian elimination to systems of polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination, manifesting as the choice to work from left to right when selecting pivot positions.
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The concept of an abstract Wiener space is mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite- dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The classical Wiener space is the prototypical example. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction.
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This is because the electromagnetic quantities are defined differently in SI and in CGS, whereas mechanical quantities are defined identically. Furthermore, within CGS, there are several plausible ways to define electromagnetic quantities, leading to different "sub-systems", including Gaussian units, "ESU", "EMU", and Lorentz–Heaviside units. Among these choices, Gaussian units are the most common today, and "CGS units" often used specifically refers to CGS-Gaussian units.
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Mallows's Cp is equivalent to AIC in the case of (Gaussian) linear regression.
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The basic, or fundamental transverse mode of a resonator is a Gaussian beam.
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Therefore, r ≤ rmax − R can be used approximately for Gaussian beams as well.
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Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. pp. 13-15 Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.
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Based on the above geometrical description, ShaferG. Shafer, "A note on Dempster's Gaussian belief functions," School of Business, University of Kansas, Lawrence, KS, Technical Report 1992. and LiuL. Liu, "A theory of Gaussian belief functions," International Journal of Approximate Reasoning, vol.
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The method builds a multi-task Gaussian process model on the data originating from different searches progressing in tandem.Bonilla, E. V., Chai, K. M., & Williams, C. (2008). Multi-task Gaussian process prediction. Advances in neural information processing systems (pp. 153-160).
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In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and quadratic costs.
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The previous solution, where the constellation has a Gaussian shape, is called constellation shaping.
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The binomial coefficient has a q-analog generalization known as the Gaussian binomial coefficient.
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In the case of the Shannon–Hartley theorem, the noise is assumed to be generated by a Gaussian process with a known variance. Since the variance of a Gaussian process is equivalent to its power, it is conventional to call this variance the noise power. Such a channel is called the Additive White Gaussian Noise channel, because Gaussian noise is added to the signal; "white" means equal amounts of noise at all frequencies within the channel bandwidth. Such noise can arise both from random sources of energy and also from coding and measurement error at the sender and receiver respectively.
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Distribution of Gaussian primes in the complex plane, with norms less than 500 The Gaussian integers are complex numbers of the form , where and are ordinary integersThe phrase "ordinary integer" is commonly used for distinguishing usual integers from Gaussian integers, and more generally from algebraic integers. and is the square root of negative one. By defining an analog of the Euclidean algorithm, Gaussian integers can be shown to be uniquely factorizable, by the argument above. Reprinted in and This unique factorization is helpful in many applications, such as deriving all Pythagorean triples or proving Fermat's theorem on sums of two squares.
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A machine-learning algorithm that involves a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution. For multi-output predictions, multivariate Gaussian processes are used, for which the multivariate Gaussian distribution is the marginal distribution at each point. For some kernel functions, matrix algebra can be used to calculate the predictions using the technique of kriging.
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With a normal distribution, differential entropy is maximized for a given variance. A Gaussian random variable has the largest entropy amongst all random variables of equal variance, or, alternatively, the maximum entropy distribution under constraints of mean and variance is the Gaussian.
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In computational chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.
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In mathematics, specifically, in measure theory, Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite- dimensional result that a Gaussian random variable has exponential tails. The result was proved in 1970 by the mathematician Xavier Fernique.
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Several types of atomic orbitals can be used: Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals. Out of the three, Gaussian-type orbitals are by far the most often used, as they allow efficient implementations of Post- Hartree–Fock methods.
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Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results.
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This broadening effect is described by a Gaussian profile and there is no associated shift.
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Clouds is the first studio album by Dutch ambient group Gaussian Curve, released in 2015.
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Typical generative model approaches include naive Bayes classifiers, Gaussian mixture models, variational autoencoders and others.
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The difference between a small and large Gaussian blur In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss). It is a widely used effect in graphics software, typically to reduce image noise and reduce detail. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination. Gaussian smoothing is also used as a pre-processing stage in computer vision algorithms in order to enhance image structures at different scales—see scale space representation and scale space implementation.
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The slow "standard algorithm" for k-means clustering, and its associated expectation- maximization algorithm, is a special case of a Gaussian mixture model, specifically, the limiting case when fixing all covariances to be diagonal, equal and have infinitesimal small variance. Instead of small variances, a hard cluster assignment can also be used to show another equivalence of k-means clustering to a special case of "hard" Gaussian mixture modelling. This does not mean that it is efficient to use Gaussian mixture modelling to compute k-means, but just that there is a theoretical relationship, and that Gaussian mixture modelling can be interpreted as a generalization of k-means; on the contrary, it has been suggested to use k-means clustering to find starting points for Gaussian mixture modelling on difficult data.
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In physics, a non-Gaussianity is the correction that modifies the expected Gaussian function estimate for the measurement of a physical quantity. In physical cosmology, the fluctuations of the cosmic microwave background are known to be approximately Gaussian, both theoretically as well as experimentally. However, most theories predict some level of non-Gaussianity in the primordial density field. Detection of these non-Gaussian signatures will allow discrimination between various models of inflation and their alternatives.
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The second edition, published in 1975, used Gaussian units exclusively, but the third edition, published in 1998, uses mostly SI units. Similarly, Electricity and Magnetism by Edward Purcell is a popular undergraduate textbook. The second edition, published in 1984, used Gaussian units, while the third edition, published in 2013, switched to SI units. The 8th SI Brochure acknowledges that the CGS-Gaussian unit system has advantages in classical and relativistic electrodynamics,, p.
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For small to moderate levels of Gaussian noise, the median filter is demonstrably better than Gaussian blur at removing noise whilst preserving edges for a given, fixed window size. However, its performance is not that much better than Gaussian blur for high levels of noise, whereas, for speckle noise and salt-and-pepper noise (impulsive noise), it is particularly effective. Because of this, median filtering is very widely used in digital image processing.
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Gaussian processes can also be used in the context of mixture of experts models, for example. The underlying rationale of such a learning framework consists in the assumption that a given mapping cannot be well captured by a single Gaussian process model. Instead, the observation space is divided into subsets, each of which is characterized by a different mapping function; each of these is learned via a different Gaussian process component in the postulated mixture.
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Fluids 18, 075104. Y. Li & C. Meneveau, “Intermittency trends and Lagrangian evolution of non-gaussian statistics in turbulent flow and scalar transport” (2006), J. Fluid Mech., 558, 133-142. Y. Li & C. Meneveau, “Origin of non-Gaussian statistics in hydrodynamic turbulence” (2005), Phys. Rev. Lett.
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The Harris affine detector relies heavily on both the Harris measure and a Gaussian scale space representation. Therefore, a brief examination of both follow. For a more exhaustive derivations see corner detection and Gaussian scale space or their associated papers.C. Harris and M. Stephens (1988).
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Conventional spatial filtering techniques for noise removal include: mean (convolution) filtering, median filtering and Gaussian smoothing.
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It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure γ on a separable Banach space E is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional L ∈ E∗ except L = 0, the push-forward measure L∗(γ) is a non- degenerate (centered) Gaussian measure on R in the sense defined above. For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.
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Functions based on the Gaussian function are natural choices, because convolution with a Gaussian gives another Gaussian whether applied to x and y or to the radius. Similarly to wavelets, another of its properties is that it is halfway between being localized in the configuration (x and y) and in the spectral (j and k) representation. As an interpolation function, a Gaussian alone seems too spread out to preserve the maximum possible detail, and thus the second derivative is added. As an example, when printing a photographic negative with plentiful processing capability and on a printer with a hexagonal pattern, there is no reason to use sinc function interpolation.
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Animation showing the deformation of a helicoid into a catenoid. The deformation is accomplished by bending without stretching. During the process, the Gaussian curvature of the surface at each point remains constant. A sphere of radius R has constant Gaussian curvature which is equal to 1/R2.
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Vecchia approximation is a Gaussian processes approximation technique originally developed by Aldo Vecchia, a statistician at United States Geological Survey. It is one of the earliest attempts to use Gaussian processes in high-dimensional settings. It has since been extensively generalized giving rise to many contemporary approximations.
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This system is still used in some subfields of physics. However, the units in that system are related to Gaussian units by factors of , which means that their magnitudes remained, like those of the Gaussian units, either far too large or far too small for practical applications.
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The Gaussian profile approximation provides an alternative means of comparison: using the approximation above shows that the RMS width \sigma of the Gaussian approximation to the Airy disk is about one-third the Airy disk radius, i.e. 0.42 \lambda N as opposed to 1.22 \lambda N.
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Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur) would more accurately reproduce the bokeh effect.
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J. Hehre and Robert J. Steward, a least squares representation of the Slater atomic orbitals as a sum of Gaussian-type orbitals is used. In their 1969 paper, the fundamentals of this principle are discussed and then further improved and used in the GAUSSIAN DFT code.
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The chi-square distribution is obtained as the sum of the squares of k independent, zero-mean, unit- variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.
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Another way to generate a signal with the required Doppler power spectrum is to pass a white Gaussian noise signal through a Gaussian filter with a frequency response equal to the square-root of the Doppler spectrum required. Although simpler than the models above, and non-deterministic, it presents some implementation questions related to needing high-order filters to approximate the irrational square-root function in the response and sampling the Gaussian waveform at an appropriate rate.
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In communication channel testing and modelling, Gaussian noise is used as additive white noise to generate additive white Gaussian noise. In telecommunications and computer networking, communication channels can be affected by wideband Gaussian noise coming from many natural sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or Johnson–Nyquist noise), shot noise, black-body radiation from the earth and other warm objects, and from celestial sources such as the Sun.
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A Gaussian blur effect is typically generated by convolving an image with an FIR kernel of Gaussian values. In practice, it is best to take advantage of the Gaussian blur’s separable property by dividing the process into two passes. In the first pass, a one-dimensional kernel is used to blur the image in only the horizontal or vertical direction. In the second pass, the same one-dimensional kernel is used to blur in the remaining direction.
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Gaussian smoothing is commonly used with edge detection. Most edge-detection algorithms are sensitive to noise; the 2-D Laplacian filter, built from a discretization of the Laplace operator, is highly sensitive to noisy environments. Using a Gaussian Blur filter before edge detection aims to reduce the level of noise in the image, which improves the result of the following edge-detection algorithm. This approach is commonly referred to as Laplacian of Gaussian, or LoG filtering.
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In mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel, is a q-analogue of the Gaussian or normal distribution. The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.
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Given large enough samples, the distribution of scores on hardiness measures will approximate a normal, Gaussian distribution.
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If stability is required in the general case, Gaussian elimination with partial pivoting (GEPP) is recommended instead.
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Thus, forecasting with Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective.
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His efforts led him to generalize the harmonic law to obtain the generalized inverse Gaussian distribution density.
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Unlike the hydrogen- like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals).
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Intensity of a simulated Gaussian beam around focus at an instant of time, showing two intensity peaks for each wavefront. Top: transverse intensity profile of a Gaussian beam that is propagating out of the page. Blue curve: electric (or magnetic) field amplitude vs. radial position from the beam axis.
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Efficient generation of fractional Brownian surfaces poses significant challenges. Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient method is Stein's method, which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process. The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter.
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Q-Chem software is maintained and distributed by Q-Chem, Inc., located in Pleasanton, California, USA. It was founded in 1993 as a result of disagreements within the Gaussian company that led to the departure (and subsequent "banning") of John Pople and a number of his students and postdocs (see Gaussian License ControversyBanned By Gaussian). The first lines of the Q-Chem code were written by Peter Gill, at that time a postdoc of Pople, during a winter vacation (December 1992) in Australia.
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In laser science, the parameter M2, also known as the beam quality factor, represents the degree of variation of a beam from an ideal Gaussian beam. It is calculated from the ratio of the beam parameter product (BPP) of the beam to that of a Gaussian beam with the same wavelength. It relates the beam divergence of a laser beam to the minimum focussed spot size that can be achieved. For a single mode TEM00 (Gaussian) laser beam, M2 is exactly one.
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The Peres–Horodecki criterion has been extended to continuous variable systems. Simon formulated a particular version of the PPT criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for 1\oplus1 -mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for 1\oplus n -mode Gaussian states, but no longer sufficient for 2\oplus2 -mode Gaussian states.
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Richer dynamic Gaussian copulas apply Monte Carlo simulation and come at the cost of requiring powerful computer technology.
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The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.
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"Costate Estimation in Optimal Control Using Integral Gaussian Quadrature Orthogonal Collocation Methods" Optimal Control Applications and Methods, 2014.
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A variant of MSK called Gaussian minimum-shift keying (GMSK) is used in the GSM mobile phone standard.
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Kernel density estimate with diagonal bandwidth for synthetic normal mixture data. We consider estimating the density of the Gaussian mixture , from 500 randomly generated points. We employ the Matlab routine for 2-dimensional data. The routine is an automatic bandwidth selection method specifically designed for a second order Gaussian kernel.
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If the εi are assumed to be i.i.d. Gaussian (with zero mean), then the model has three parameters: b0, b1, and the variance of the Gaussian distributions. Thus, when calculating the AIC value of this model, we should use k=3. More generally, for any least squares model with i.i.d.
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Lower-end digital cameras, including many mobile phone cameras, commonly use gaussian blurring to cover up image noise caused by higher ISO light sensitivities. Gaussian blur automatically is applied as part of the image post-processing of the photo by the camera software, leading to an irreversible loss of detail.
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Linnik obtained numerous results concerning infinitely divisible distributions. In particular, he proved the following generalisation of Cramér's theorem: any divisor of a convolution of Gaussian and Poisson random variables is also a convolution of Gaussian and Poisson. He has also coauthored the book on the arithmetics of infinitely divisible distributions.
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For general stochastic processes strict-sense stationarity implies wide-sense stationarity but not every wide-sense stationary stochastic process is strict-sense stationary. However, for a Gaussian stochastic process the two concepts are equivalent. A Gaussian stochastic process is strict-sense stationary if, and only if, it is wide- sense stationary.
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1966, 5239;H. Nozaki, H. Takaya, S. Moriuti, R. Noyori, Tetrahedron 1968, 24, 3655. ;1970: John Pople develops the Gaussian program greatly easing computational chemistry calculations.W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Gaussian 70 (Quantum Chemistry Program Exchange, Program No. 237, 1970).
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If jitter has a Gaussian distribution, it is usually quantified using the standard deviation of this distribution. This translates to an RMS measurement for a zero-mean distribution. Often, jitter distribution is significantly non-Gaussian. This can occur if the jitter is caused by external sources such as power supply noise.
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Nowadays, process data can be much more complex, e.g. non-Gaussian, mix numerical and categorical, or be missing- valued.
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In machine learning and statistics, EM (expectation maximization) algorithm handles latent variables, while GMM is the Gaussian mixture model.
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P3P methods assume that the data is noise free, most PnP methods assume Gaussian noise on the inlier set.
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Indeed, according to , the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.
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Later, digital filters replaced analog filters and international standards such as ISO 11562 for the Gaussian filter were published.
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In mathematics, his name is frequently attached to an efficient Gaussian elimination method for tridiagonal matrices—the Thomas algorithm.
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43,112,609 is not a Gaussian prime, the largest of only 28 known Mersenne prime indexes to have this property.
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The natural Gaussian primes are exactly the natural primes ending with 7 or (i.e. the natural primes congruent to ).
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In color cameras where more amplification is used in the blue color channel than in the green or red channel, there can be more noise in the blue channel. At higher exposures, however, image sensor noise is dominated by shot noise, which is not Gaussian and not independent of signal intensity. Also, there are many Gaussian denoising algorithms.Mehdi Mafi, Harold Martin, Jean Andrian, Armando Barreto, Mercedes Cabrerizo, Malek Adjouadi, “A Comprehensive Survey on Impulse and Gaussian Denoising Filters for Digital Images,” Signal Processing, vol.
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Since side lobes of the Airy disk are responsible for degrading the image, techniques for suppressing them are utilized. In case the imaging beam has Gaussian distribution, when the truncation ratio (the ratio of the diameter of the Gaussian beam to the diameter of the truncating aperture) is set to 1, the side-lobes become negligible and the beam profile becomes purely Gaussian. The measured beam profile of such imaging system is shown and compared to the modeled beam profile in the Figure on the right.
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This can also be shown with the continuous Fourier transform, as follows. The Fourier transform analyzes a signal in terms of its frequencies, transforms convolutions into products, and transforms Gaussians into Gaussians. The Weierstrass transform is convolution with a Gaussian and is therefore multiplication of the Fourier transformed signal with a Gaussian, followed by application of the inverse Fourier transform. This multiplication with a Gaussian in frequency space blends out high frequencies, which is another way of describing the "smoothing" property of the Weierstrass transform.
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One method to remove noise is by convolving the original image with a mask that represents a low-pass filter or smoothing operation. For example, the Gaussian mask comprises elements determined by a Gaussian function. This convolution brings the value of each pixel into closer harmony with the values of its neighbors. In general, a smoothing filter sets each pixel to the average value, or a weighted average, of itself and its nearby neighbors; the Gaussian filter is just one possible set of weights.
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In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for 1\oplus1 -mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for 1\oplus n -mode Gaussian states, but no longer sufficient for 2\oplus2 -mode Gaussian states.
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There are several common parametric empirical Bayes models, including the Poisson–gamma model (below), the Beta- binomial model, the Gaussian–Gaussian model, the Dirichlet-multinomial model, as well specific models for Bayesian linear regression (see below) and Bayesian multivariate linear regression. More advanced approaches include hierarchical Bayes models and Bayesian mixture models.
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Within a study, values obtained by close Gaussian kernels are summed, though values are combined by square- distance-weighted averaging.
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It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature.
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The fractional Brownian motion is a Gaussian process whose covariance function is a generalisation of that of the Wiener process.
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It has also been compared to the natural evolution of populations of living organisms. In this case s(x) is the probability that the individual having an array x of phenotypes will survive by giving offspring to the next generation; a definition of individual fitness given by Hartl 1981. The yield, P, is replaced by the mean fitness determined as a mean over the set of individuals in a large population. Phenotypes are often Gaussian distributed in a large population and a necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation, with respect to all Gaussian quantitative characters, is that it may push the centre of gravity of the Gaussian to the centre of gravity of the selected individuals.
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Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature.Weisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets.
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The POLYATOM SystemI.G. Csizmadia, M.C. Harrison, J.W. Moskowitz and B.T. Sutcliffe, Nonempirical LCAO-MO-SCF-CI calculations on organic molecules with gaussian-type functions. Introductory review and mathematical formalism, Theoretica Chimica Acta, 6, 191, 1966. was the first package for ab initio calculations using Gaussian orbitals that was applied to a wide variety of molecules.
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Some evidence for this can be had by Monte Carlo simulations. (PDF 1465 kB) The key approximation property used to construct these filters is that the state prediction density is approximately Gaussian. Masreliez discovered in 1975 that this approximation yields an intuitively appealing non-Gaussian filter recursions, with data dependent covariance (unlike the Gaussian case) this derivation also provides one of the nicest ways of establishing the standard Kalman filter recursions. Some theoretical justification for use of the Masreliez approximation is provided by the "continuity of state prediction densities" theorem in Martin (1979).
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Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the Gaussian curvature is zero and the integral of Gaussian curvature at a vertex is equal to the defect there. This can be used to calculate the number V of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect. This total will have one complete circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron.
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The resulting effect is the same as convolving with a two- dimensional kernel in a single pass, but requires fewer calculations. Discretization is typically achieved by sampling the Gaussian filter kernel at discrete points, normally at positions corresponding to the midpoints of each pixel. This reduces the computational cost but, for very small filter kernels, point sampling the Gaussian function with very few samples leads to a large error. In these cases, accuracy is maintained (at a slight computational cost) by integration of the Gaussian function over each pixel's area.
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One of the computational advantages of the Bellman pseudospectral method is that it allows one to escape Gaussian rules in the distribution of node points. That is, in a standard pseudospectral method, the distribution of node points are Gaussian (typically Gauss-Lobatto for finite horizon and Gauss-Radau for infinite horizon). The Gaussian points are sparse in the middle of the interval (middle is defined in a shifted sense for infinite- horizon problems) and dense at the boundaries. The second-order accumulation of points near the boundaries have the effect of wasting nodes.
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From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor.
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Real laser beams are often non-Gaussian, being multi- mode or mixed-mode. Multi-mode beam propagation is often modeled by considering a so-called "embedded" Gaussian, whose beam waist is M times smaller than that of the multimode beam. The diameter of the multimode beam is then M times that of the embedded Gaussian beam everywhere, and the divergence is M times greater, but the wavefront curvature is the same. The multimode beam has M2 times the beam area but 1/M2 less beam intensity than the embedded beam.
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Gaussian adaptation has also been used for other purposes as for instance shadow removal by "The Stauffer-Grimson algorithm" which is equivalent to Gaussian adaptation as used in the section "Computer simulation of Gaussian adaptation" above. In both cases the maximum likelihood method is used for estimation of mean values by adaptation at one sample at a time. But there are differences. In the Stauffer-Grimson case the information is not used for the control of a random number generator for centering, maximization of mean fitness, average information or manufacturing yield.
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A simple example is the rational prime 5, which is factored as in the table, and therefore not a Gaussian prime.
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DBSCAN can find non-linearly separable clusters. This dataset cannot be adequately clustered with k-means or Gaussian Mixture EM clustering.
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The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general topological vector space.
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The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
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First, generally equity returns do not follow a Gaussian distribution and their third and fourth moments confirm skewed and leptokurtic distributions.
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These all have positive Gaussian curvature. The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature. The degenerate form :X_0^2-X_1^2-X_2^2=0.
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The distribution is extremely spiky and leptokurtic, reason why researchers had to turn their backs to statistics to solve e.g. authorship attribution problems. Nevertheless, usage of Gaussian statistics is perfectly possible by applying data transformation. Van Droogenbroeck F.J., 'An essential rephrasing of the Zipf-Mandelbrot law to solve authorship attribution applications by Gaussian statistics' (2019) 3\.
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Gaussian residuals, the variance of the residuals' distributions should be counted as one of the parameters. As another example, consider a first-order autoregressive model, defined by xi = c \+ φxi−1 \+ εi, with the εi being i.i.d. Gaussian (with zero mean). For this model, there are three parameters: c, φ, and the variance of the εi.
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NCEP T62 Gaussian grid points A Gaussian grid is used in the earth sciences 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 each latitude (or parallel), i.e.
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Despite the simple formula for the probability density function, numerical probability calculations for the inverse Gaussian distribution nevertheless require special care to achieve full machine accuracy in floating point arithmetic for all parameter values. Functions for the inverse Gaussian distribution are provided for the R programming language by several packages including rmutil, SuppDists, STAR, invGauss, LaplacesDemon, and statmod.
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Gaussian states are a paradigmatic class of states of continuous variable quantum systems. Although they can nowadays be created and manipulated in, e.g, state-of-the-art optical platforms, naturally robust to decoherence, it is well-known that they are not sufficient for, e.g., universal quantum computing because transformations that preserve the Gaussian nature of a state are linear.
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Gaussian blur is a low-pass filter, attenuating high frequency signals.R.A. Haddad and A.N. Akansu, "A Class of Fast Gaussian Binomial Filters for Speech and Image Processing," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 39, pp 723-727, March 1991. Its amplitude Bode plot (the log scale in the frequency domain) is a parabola.
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Features of Calculation and Design of High-Speed Radio Links for Earth Remote Sensing Spacecraft. Moreover, a careful design of the constellation geometry can approach the Gaussian capacity as the constellation size grows to infinity. For the regular QAM constellations, a gap of 1.56 dB is observed.H. Méric, Approaching The Gaussian Channel Capacity With APSK Constellations, IEEE Communications Letters.
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The Gaussian year is the sidereal year for a planet of negligible mass (relative to the Sun) and unperturbed by other planets that is governed by the Gaussian gravitational constant. Such a planet would be slightly closer to the Sun than Earth's mean distance. Its length is: : days (365 d 6 h 9 min 56 s).
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With temporal solitons it is possible to remove such a problem completely. linear and nonlinear effects on Gaussian pulses Consider the picture on the right. On the left there is a standard Gaussian pulse, that's the envelope of the field oscillating at a defined frequency. We assume that the frequency remains perfectly constant during the pulse.
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The GIG distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture.Dimitris Karlis, "An EM type algorithm for maximum likelihood estimation of the normal–inverse Gaussian distribution", Statistics & Probability Letters 57 (2002) 43–52.Barndorf-Nielsen, O.E., 1997. Normal Inverse Gaussian Distributions and stochastic volatility modelling. Scand.
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Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the Gaussian basis. Since then much work has been done to speed up the evaluation of these integrals which are the slowest part of many quantum chemical calculations. Živković and Maksić (1968) suggested using Hermite Gaussian functions, as this simplifies the equations.
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The confined Gaussian window yields the smallest possible root mean square frequency width for a given temporal width . These windows optimize the RMS time-frequency bandwidth products. They are computed as the minimum eigenvectors of a parameter-dependent matrix. The confined Gaussian window family contains the and the in the limiting cases of large and small , respectively.
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Depending on the chosen mode, different amounts of band- pass Gaussian noise are added to the synthesized harmonic signal by the decoder.
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Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium. However the speed of transmitting information is always limited to c.
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The stochastic vacuum model is based on the approximation of nonperturbative QCD as a Gaussian process. It allows to calculate Wilson loops.
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The resulting metric makes the open Möbius band into a (geodesically) complete flat surface (i.e., having Gaussian curvature equal to 0 everywhere).
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Wojciech Jarosz. 2001. Fast Image Convolutions. Box blurs are frequently used to approximate a Gaussian blur.W3C SVG1.1 specification, 15.17 Filter primitive 'feGaussianBlur'.
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The use of Gaussian orbitals in electronic structure theory (instead of the more physical Slater-type orbitals) was first proposed by Boys in 1950. The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4—5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.
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Laguerre-Gaussian beams also possess a well-defined orbital angular momentum that can rotate particles.Curtis JE, Grier DG, "Structure of Optical Vortices" (2003).
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In technical terms, the distribution of price changes is leptokurtic, with more extreme events in the tails than are found in Gaussian distribution.
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For small values of ε (10−6 to 10−8) the errors introduced by truncating the Gaussian are usually negligible. For larger values of ε, however, there are many better alternatives to a rectangular window function. For example, for a given number of points, a Hamming window, Blackman window, or Kaiser window will do less damage to the spectral and other properties of the Gaussian than a simple truncation will. Notwithstanding this, since the Gaussian kernel decreases rapidly at the tails, the main recommendation is still to use a sufficiently small value of ε such that the truncation effects are no longer important.
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An alternative approach is to use the discrete Gaussian kernel:Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254. :T(n, t) = e^{-t} I_n(t) where I_n(t) denotes the modified Bessel functions of integer order. This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.Campbell, J, 2007, The SMM model as a boundary value problem using the discrete diffusion equation, Theor Popul Biol.
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GPOPS-II (pronounced "GPOPS 2") is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming. The acronym GPOPS stands for "General Purpose OPtimal Control Software", and the Roman numeral "II" refers to the fact that GPOPS-II is the second software of its type (that employs Gaussian quadrature integration).
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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as . This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.
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When implementing scale-space smoothing in practice there are a number of different approaches that can be taken in terms of continuous or discrete Gaussian smoothing, implementation in the Fourier domain, in terms of pyramids based on binomial filters that approximate the Gaussian or using recursive filters. More details about this are given in a separate article on scale space implementation.
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The diffusion equation is continuous in both space and time. One may discretize space, time, or both space and time, which arise in application. Discretizing time alone just corresponds to taking time slices of the continuous system, and no new phenomena arise. In discretizing space alone, the Green's function becomes the discrete Gaussian kernel, rather than the continuous Gaussian kernel.
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In addition, he later became head of the Gaussian Border Office of the Gaussian Bavarian Ostmark of the NSDAP. From April 1937 he directed a real school in Neuburg on the Danube. Concurrently, he was still the regional leader of the Bavarian State Parliament of the Federal Republic of Germany. He died in September 1939 in Neuburg on the Danube.
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Gaussian convolutions are used extensively in signal and image processing. For example, image-blurring can be accomplished with Gaussian convolution where the \sigma parameter will control the strength of the blurring. Higher values would thus correspond to a more blurry end result. It is also commonly used in Computer vision applications such as Scale-invariant feature transform (SIFT) feature detection.
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The two-gaussian model parameters, including the development process, can be determined experimentally by exposing shapes for which the Gaussian integral is easily solved, i.e. donuts, with increasing dose and observing at which dose the center resist clears or does not clear. A thin resist with a low electron density will reduce forward scattering. A light substrate (light nuclei) will reduce backscattering.
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The notion of a kernel plays a crucial role in Bayesian probability as the covariance function of a stochastic process called the Gaussian process.
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Position space probability density of an initially Gaussian state trapped in an infinite potential well experiencing periodic Quantum Tunneling in a centered potential wall.
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In computer vision and image processing, the notion of scale space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.
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The noncentral chi-square distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.
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In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale (see the article on scale space). A special type of scale-space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale-space axioms).
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Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp.
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PCA of a multivariate Gaussian distribution. The vectors shown are the first (longer vector) and second principal components, which indicate the directions of maximum variance.
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The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/(2μ2) and −λ/2, and natural statistics X and 1/X.
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Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Julius Springer Verlag, 1926. It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss proved the method under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian).
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Some lasers, particularly high- power ones, produce multimode beams, with the transverse modes often approximated using Hermite–Gaussian or Laguerre-Gaussian functions. Some high power lasers use a flat-topped profile known as a "tophat beam". Unstable laser resonators (not used in most lasers) produce fractal-shaped beams. Specialized optical systems can produce more complex beam geometries, such as Bessel beams and optical vortexes.
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As in the Gaussian units, the Heaviside–Lorentz units (HLU in this article) use the length–mass–time dimensions. This means that all of the electric and magnetic units are expressible in terms of the base units of length, time and mass. Coulomb's equation, used to define charge in these systems, is in the Gaussian system, and in the HLU. The unit of charge then connects to .
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Note that Gaussian quadrature can also be adapted for various weight functions, but the technique is somewhat different. In Clenshaw–Curtis quadrature, the integrand is always evaluated at the same set of points regardless of w(x), corresponding to the extrema or roots of a Chebyshev polynomial. In Gaussian quadrature, different weight functions lead to different orthogonal polynomials, and thus different roots where the integrand is evaluated.
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In statistics, originally in geostatistics, kriging or Gaussian process regression is a method of interpolation for which the interpolated values are modeled by a Gaussian process governed by prior covariances. Under suitable assumptions on the priors, kriging gives the best linear unbiased prediction of the intermediate values. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the most likely intermediate values.
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In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.
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Gaussian caustic means that each micro-surface obey gaussian distribution. The position and orientation of each of the micro-surfaces is then obtained using a combination of Poisson integration and simulated annealing. There have been many different approaches to address the continuous problem. One approach uses an idea from transportation theory called optimal transport to find a mapping between incoming light rays and the target surface.
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A Gaussian quadrature rule is typically more accurate than a Newton–Cotes rule, which requires the same number of function evaluations, if the integrand is smooth (i.e., if it is sufficiently differentiable). Other quadrature methods with varying intervals include Clenshaw–Curtis quadrature (also called Fejér quadrature) methods, which do nest. Gaussian quadrature rules do not nest, but the related Gauss–Kronrod quadrature formulas do.
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If \Delta_4>0, and the surface has real points, it is either a hyperbolic paraboloid or a one-sheet hyperboloid. In both cases, this is a ruled surface that has a negative Gaussian curvature at every point. If \Delta_4<0, the surface is either an ellipsoid or a two-sheet hyperboloid or an elliptic paraboloid. In all cases, it has a positive Gaussian curvature at every point.
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Quantum clustering (QC), is a data clustering algorithm accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a sigma value, a hyper-parameter which can be manually defined and manipulated to suit the application. Gradient descent is then used to "move" the points to their local minima. These local minima then define the cluster centers.
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The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle. The Gaussian correlation inequality (GCI), formerly known as the Gaussian correlation conjecture (GCC), is a mathematical theorem in the fields of mathematical statistics and convex geometry. A special case of the inequality was published as a conjecture in a paper from 1955;Dunnett, C. W.; Sobel, M. Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution. Biometrika 42, (1955). 258–260.
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T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals. Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100.
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In practice, when computing a discrete approximation of the Gaussian function, pixels at a distance of more than 3σ have a small enough influence to be considered effectively zero. Thus contributions from pixels outside that range can be ignored. Typically, an image processing program need only calculate a matrix with dimensions \lceil6\sigma\rceil × \lceil6\sigma\rceil (where \lceil \cdot \rceil is the ceiling function) to ensure a result sufficiently 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 calculations, and so is termed separable filter.
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Together with Christoph Adami, he defined the quantum version of conditional and mutual entropies, which are basic notions of Shannon's information theory, and discovered that quantum information can be negative (a pair of entangled particles was coined a qubit-antiqubit pair). This has led to important results in quantum information sciences, for example quantum state merging. He is best known today for his work on quantum information with continuous variables. He found a Gaussian quantum cloning transformation (see no-cloning theorem) and invented a Gaussian quantum key distribution protocol, which is the continuous counterpart of the so-called BB84 protocol, making a link with Shannon's theory of Gaussian channels.
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M2 is useful because it reflects how well a collimated laser beam can be focused to a small spot, or how well a divergent laser source can be collimated. It is a better guide to beam quality than Gaussian appearance because there are many cases in which a beam can look Gaussian, yet have an M2 value far from unity. Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California Likewise, a beam intensity profile can appear very "un-Gaussian", yet have an M2 value close to unity. The value of M2 is determined by measuring D4σ or "second moment" width.
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In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version can be defined on any graph, usually a lattice in d-dimensional Euclidean space. The continuum version is defined on Rd or on a bounded subdomain of Rd. It can be thought of as a natural generalization of one-dimensional Brownian motion to d time (but still one space) dimensions; in particular, the one-dimensional continuum GFF is just the standard one- dimensional Brownian motion or Brownian bridge on an interval.
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The retention time is the time from the start of signal detection to the time of the peak height of the Gaussian curve. From the variables in the figure above, the resolution, plate number, and plate height of the column plate model can be calculated using the equations: Resolution (Rs) Rs = 2(tRB – tRA)/(wB \+ wA) Where: tRB = retention time of solute B tRA = retention time of solute A wB = Gaussian curve width of solute B wA = Gaussian curve width of solute A Plate Number (N): N = (tR)2/(w/4)2 Plate Height (H): H = L/N Where L is the length of the column.
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In maximum likelihood beamformer (DML), the noise is modeled as a stationary Gaussian white random processes while the signal waveform as deterministic (but arbitrary) and unknown.
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The Gaussian molecular orbital methods were described in the 1986 book Ab initio molecular orbital theory by Warren Hehre, Leo Radom, Paul v.R. Schleyer and Pople.
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Cramér's theorem implies that a linear combination of independent non- Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.
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Let \xi and \eta are independent random variables. If \xi+\eta and \xi-\eta are independent then \xi and \eta have normal distributions (the Gaussian distribution).
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This proof was simplified by Gauss in his Disquisitiones Arithmeticae (art. 182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers.
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Gaussian charts are often less convenient than Schwarzschild or isotropic charts. However, they have found occasional application in the theory of static spherically symmetric perfect fluids.
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The above is for SI units. In some cases, the cyclotron frequency is given in Gaussian units.Kittel, Charles. Introduction to Solid State Physics, 8th edition. pp.
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The two-piece normal, binormal, or double Gaussian distribution: its origin and rediscoveries. Statistical Science, vol. 29, no. 1, pp.106-112. doi:10.1214/13-STS417.
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In probability theory, Dudley's theorem is a result relating the expected upper bound and regularity properties of a Gaussian process to its entropy and covariance structure.
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The Gaussian periods are related to the Gauss sums G(1,\chi) for which the character χ is trivial on H. Such χ take the same value at all elements a in a fixed coset of H in G. For example, the quadratic character mod p described above takes the value 1 at each quadratic residue, and takes the value -1 at each quadratic non-residue. The Gauss sum G(1,\chi) can thus be written as a linear combination of Gaussian periods (with coefficients χ(a)); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ)×. In other words, the Gaussian periods and Gauss sums are each other's Fourier transforms. The Gaussian periods generally lie in smaller fields, since for example when n is a prime p, the values χ(a) are (p − 1)-th roots of unity.
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482-489, Nov. 2013. Sotirios P. Chatzis, Yiannis Demiris, “Nonparametric mixtures of Gaussian processes with power-law behaviour,” IEEE Transactions on Neural Networks and Learning Systems, vol.
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The reconstruction is also called 'quantum tomographic reconstruction'. For squeezed states, the Wigner function has a Gaussian shape, with an elliptical contour line, see Fig.: 1(f).
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He was a member of the Society for Industrial and Applied Mathematics (SIAM) student chapter. His work on Gaussian matrices was awarded the SIAM best student paper.
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Form the initial sample set and weights by sampling according to the prior distribution. For example, specify as Gaussian and set the weights equal to each other.
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Bluetooth Special Interest Group. Retrieved from Bluetooth Core Specifications, 1 December 2017. Page 2535. Originally, Gaussian frequency-shift keying (GFSK) modulation was the only modulation scheme available.
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Gaussian Curve is an ambient group formed in Amsterdam in 2014. It consists of Italian musician Gigi Masin, Scottish musician Jonny Nash and Dutch musician Marco Sterk.
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For a Gaussian distribution, this is the best unbiased estimator (i.e., one with the lowest MSE among all unbiased estimators), but not, say, for a uniform distribution.
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A measurement of dynamical heterogeneity can be done by calculating correlation functions like Non- Gaussian parameter, four point correlation functions (Dynamic Susceptibility) and three time correlation functions.
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This property is related to the Heisenberg uncertainty principle, but not directly – see Gabor limit for discussion. The product of the standard deviation in time and frequency is limited. The boundary of the uncertainty principle (best simultaneous resolution of both) is reached with a Gaussian window function, as the Gaussian minimizes the Fourier uncertainty principle. This is called the Gabor transform (and with modifications for multiresolution becomes the Morlet wavelet transform).
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Principal sources of Gaussian noise in digital images arise during acquisition e.g. sensor noise caused by poor illumination and/or high temperature, and/or transmission e.g. electronic circuit noise. In digital image processing Gaussian noise can be reduced using a spatial filter, though when smoothing an image, an undesirable outcome may result in the blurring of fine-scaled image edges and details because they also correspond to blocked high frequencies.
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Weierstrass used this transform in his original proof of the Weierstrass approximation theorem. It is also known as the Gauss transform or Gauss–Weierstrass transform after Carl Friedrich Gauss and as the Hille transform after Einar Carl Hille who studied it extensively. The generalization Wt mentioned below is known in signal analysis as a Gaussian filter and in image processing (when implemented on R2) as a Gaussian blur.
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In Gaussian units, unlike SI units, the electric field E and the magnetic field B have the same dimension. This amounts to a factor of c between how B is defined in the two unit systems, on top of the other differences. (The same factor applies to other magnetic quantities such as H and M.) For example, in a planar light wave in vacuum, in Gaussian units, while in SI units.
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In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the crosscorrelation of a Gaussian signal before and after it has passed through a nonlinear operation are equal up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the Massachusetts Institute of Technology.J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec.
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The Gaussian filter is non-causal which means the filter window is symmetric about the origin in the time- domain. This makes the Gaussian filter physically unrealizable. This is usually of no consequence for applications where the filter bandwidth is much larger than the signal. In real-time systems, a delay is incurred because incoming samples need to fill the filter window before the filter can be applied to the signal.
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Gauss's Theorema Egregium, the "Remarkable Theorem", shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any isometric embedding in and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature. This theorem can expressed in terms of the power series expansion of the metric, , is given in normal coordinates as :.
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A triangulation of the torus On a sphere or a hyperboloid, the area of a geodesic triangle, i.e. a triangle all the sides of which are geodesics, is proportional to the difference of the sum of the interior angles and . The constant of proportionality is just the Gaussian curvature, a constant for these surfaces. For the torus, the difference is zero, reflecting the fact that its Gaussian curvature is zero.
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Gaussian blurring is commonly used when reducing the size of an image. When downsampling an image, it is common to apply a low-pass filter to the image prior to resampling. This is to ensure that spurious high-frequency information does not appear in the downsampled image (aliasing). Gaussian blurs have nice properties, such as having no sharp edges, and thus do not introduce ringing into the filtered image.
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This may be accomplished by the Hardy–Weinberg law. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability independent of the structure (Kjellström, 1996). In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense Gaussian adaptation may be seen as a genetic algorithm.
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Originally available through the Quantum Chemistry Program Exchange, it was later licensed out of Carnegie Mellon University, and since 1987 has been developed and licensed by Gaussian, Inc.
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Corrections for absorption by Gaussian quadrature integration were applied. Corrections for Lorentz, polarization, and background effects were also applied as well as reduction of intensities to structure factors.
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Gaussian Process Regression Models for Predicting Water Retention Curves - Application of Machine Learning Techniques for Modelling Uncertainty in Hydraulic Curves. Retrieved from the Delft University of Technology repository.
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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.
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Pople pioneered the development of more sophisticated computational methods, called ab initio quantum chemistry methods, that use basis sets of either Slater type orbitals or Gaussian orbitals to model the wave function. While in the early days these calculations were extremely expensive to perform, the advent of high speed microprocessors has made them much more feasible today. He was instrumental in the development of one of the most widely used computational chemistry packages, the Gaussian suite of programs, including coauthorship of the first version, Gaussian 70.Gaussian's page on John Pople One of his most important original contributions is the concept of a model chemistry whereby a method is rigorously evaluated across a range of molecules.
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In Gaussian noise, each pixel in the image will be changed from its original value by a (usually) small amount. A histogram, a plot of the amount of distortion of a pixel value against the frequency with which it occurs, shows a normal distribution of noise. While other distributions are possible, the Gaussian (normal) distribution is usually a good model, due to the central limit theorem that says that the sum of different noises tends to approach a Gaussian distribution. In either case, the noise at different pixels can be either correlated or uncorrelated; in many cases, noise values at different pixels are modeled as being independent and identically distributed, and hence uncorrelated.
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In a stationary Gaussian time series model, the likelihood function is (as usual in Gaussian models) a function of the associated mean and covariance parameters. With a large number (N) of observations, the (N \times N) covariance matrix may become very large, making computations very costly in practice. However, due to stationarity, the covariance matrix has a rather simple structure, and by using an approximation, computations may be simplified considerably (from O(N^2) to O(N\log(N))). The idea effectively boils down to assuming a heteroscedastic zero-mean Gaussian model in Fourier domain; the model formulation is based on the time series' discrete Fourier transform and its power spectral density.
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The equations below assume a beam with a circular cross-section at all values of ; this can be seen by noting that a single transverse dimension, , appears. Beams with elliptical cross-sections, or with waists at different positions in for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of and of the location for the two transverse dimensions and . Arbitrary solutions of the paraxial Helmholtz equation can be expressed as combinations of Hermite–Gaussian modes (whose amplitude profiles are separable in and using Cartesian coordinates) or similarly as combinations of Laguerre–Gaussian modes (whose amplitude profiles are separable in and using cylindrical coordinates).Siegman, p. 642.
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133 is a semiprime: a product of two prime numbers, namely 7 and 19. Since those prime factors are Gaussian primes, this means that 133 is a Blum integer.
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As a Riemannian manifold, the complex projective plane is a 4-dimensional manifold whose sectional curvature is quarter-pinched. The rival normalisations are for the curvature to be pinched between 1/4 and 1; alternatively, between 1 and 4. With respect to the former normalisation, the imbedded surface defined by the complex projective line has Gaussian curvature 1. With respect to the latter normalisation, the imbedded real projective plane has Gaussian curvature 1.
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In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): :Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.
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In other words, the random variable X is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is marginalized out (integrated out). The reason for the usefulness of this characterization is that the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian distribution. As a result, the non- standardized Student's t-distribution arises naturally in many Bayesian inference problems.
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The Ross–Fahroo methods are based on shifted Gaussian pseudospectral node points. The shifts are obtained by means of a linear or nonlinear transformation while the Gaussian pseudospectral points are chosen from a collection of Gauss- Lobatto or Gauss-Radau distribution arising from Legendre or Chebyshev polynomials. The Gauss-Lobatto pseudospectral points are used for finite- horizon optimal control problems while the Gauss-Radau pseudospectral points are used for infinite-horizon optimal control problems.
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The Tweedie convergence theorem thus provides an alternative explanation for the origin of 1/f noise, based its central limit-like effect. Much as the central limit theorem requires certain kinds of random processes to have as a focus of their convergence the Gaussian distribution and thus express white noise, the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express 1/f noise.
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It is often incorrectly assumed that Gaussian noise (i.e., noise with a Gaussian amplitude distributionsee normal distribution) necessarily refers to white noise, yet neither property implies the other. Gaussianity refers to the probability distribution with respect to the value, in this context the probability of the signal falling within any particular range of amplitudes, while the term 'white' refers to the way the signal power is distributed (i.e., independently) over time or among frequencies.
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This is because the signal mixtures share the same source signals. #Normality: According to the Central Limit Theorem, the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution. Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables. Here we consider the value of each signal as the random variable.
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Most laser beam outputs usually have Gaussian energy distribution. Using beam homogenizer will create an evenly distributed energy of the beam instead of the Gaussian shape. Unlike beam shaper who create a certain shape to the beam, beam homogenizer spread the central concentrated energy among the beam diameter so the results are sometimes grainy. An example for simple beam homogenizer can be just a murky glass, after it, the beam will be more homogenized.
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436–439, New York, NY, USA, April–May 2002. To increase the accuracy of fingerprinting methods, statistical post-processing techniques (like Gaussian process theory) can be applied, to transform discrete set of "fingerprints" to a continuous distribution of RSSI of each access point over entire location Golovan A. A. et al. Efficient localization using different mean offset models in Gaussian processes //2014 International Conference on Indoor Positioning and Indoor Navigation (IPIN). – IEEE, 2014.
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Neural Network Gaussian Processes (NNGPs) are equivalent to Bayesian neural networks in a particular limit, and provide a closed form way to evaluate Bayesian neural networks. They are a Gaussian process probability distribution which describes the distribution over predictions made by the corresponding Bayesian neural network. Computation in artificial neural networks is usually organized into sequential layers of artificial neurons. The number of neurons in a layer is called the layer width.
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For a Gaussian process, continuity in probability is equivalent to mean-square continuity, and continuity with probability one is equivalent to sample continuity. The latter implies, but is not implied by, continuity in probability. Continuity in probability holds if and only if the mean and autocovariance are continuous functions. In contrast, sample continuity was challenging even for stationary Gaussian processes (as probably noted first by Andrey Kolmogorov), and more challenging for more general processes.
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When bivariate Gaussian copulas are assigned to edges of a vine, then the resulting multivariate density is the Gaussian density parametrized by a partial correlation vine rather than by a correlation matrix. The vine pair-copula construction, based on the sequential mixing of conditional distributions has been adapted to discrete variables and mixed discrete/continuous response . Also factor copulas, where latent variables have been added to the vine, have been proposed (e.g., ).
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They wrote a new program called TEXAS based on the original MOLPRO and replaced Gaussian lobe functions with the standard Gaussian functions. TEXAS emphasized large molecules, SCF convergence, geometry optimization techniques, and vibrational spectroscopy-related calculations. From 1982 onward, the program was further developed at the University of Arkansas. The primary significant expansion was the usage of a few new electron correlation methods by Saebo and a first- order MC-SCF program by Hamilton.
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Example of application of the Rudin et al. total variation denoising technique to an image corrupted by Gaussian noise. This example created using demo_tv.m by Guy Gilboa, see external links.
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The geometric mean filter is most widely used to filter out Gaussian noise. In general it will help smooth the image with less data loss than an arithmetic mean filter.
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A symmetric distribution which can model both tail (long and short) and center behavior (like flat, triangular or Gaussian) completely independently could be derived e.g. by using X = IH/chi.
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Order: 214 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 13 ⋅ 29 = 145926144000 Schur multiplier: Order 2. Outer automorphism group: Trivial. Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers.
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Then denoising algorithms designed for the framework of additive white Gaussian noise are used; the final estimate is then obtained by applying an inverse Anscombe transformation to the denoised data.
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Apodized gratings offer significant improvement in side-lobe suppression while maintaining reflectivity and a narrow bandwidth. The two functions typically used to apodize a FBG are Gaussian and raised-cosine.
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Nonlinear and non Gaussian particle filters applied to inertial platform repositioning. LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September (1991).
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This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.
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The resulting posterior distribution is also Gaussian, with a mean and covariance that can be simply computed from the observed values, their variance, and the kernel matrix derived from the prior.
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Weingarten, Y. Steinberg, and S. Shamai, The capacity region of the Gaussian multiple-input multiple-output broadcast channel , IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 3936–3964, 2006.
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Polarized QM/MM calculations for excited states are provided in the Thole framework. It features interfaces to more external Quantum Chemistry packages (Gaussian, NWChem and ORCA) for large scale production runs.
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The canonical Gaussian cylinder set measure on an infinite-dimensional Hilbert space can never be a bona fide measure; equivalently, the identity function on such a space cannot be γ-radonifying.
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For stationary Gaussian random signals, this lower bound is usually attained at a sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimal quantization.
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For this, one has to generate two-mode entangled Gaussian states and apply a Haar-random unitary U to their "right halves", while doing nothing to the others. Then we can measure the "left halves" to find out which of the input states contained a photon before we applied U. This is precisely equivalent to scattershot boson sampling, except for the fact that our measurement of the herald photons has been deferred till the end of the experiment, instead of happening at the beginning. Therefore, approximate Gaussian boson sampling can be argued to be hard under precisely the same complexity assumption as can approximate ordinary or scattershot boson sampling. Gaussian resources can be employed at the measurement stage, as well.
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The Gaussian process emulator model treats the problem from the viewpoint of Bayesian statistics. In this approach, even though the output of the simulation model is fixed for any given set of inputs, the actual outputs are unknown unless the computer model is run and hence can be made the subject of a Bayesian analysis. The main element of the Gaussian process emulator model is that it models the outputs as a Gaussian process on a space that is defined by the model inputs. The model includes a description of the correlation or covariance of the outputs, which enables the model to encompass the idea that differences in the output will be small if there are only small differences in the inputs.
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Like Gaussian quadrature, Tanh-Sinh quadrature is well suited for arbitrary-precision integration, where an accuracy of hundreds or even thousands of digits is desired. The convergence is exponential (in the discretization sense) for sufficiently well-behaved integrands: doubling the number of evaluation points roughly doubles the number of correct digits. Tanh-Sinh quadrature is not as efficient as Gaussian quadrature for smooth integrands, but unlike Gaussian quadrature, tends to work equally well with integrands having singularities or infinite derivatives at one or both endpoints of the integration interval as already noted. Furthermore, Tanh-Sinh quadrature can be implemented in a progressive manner, with the step size halved each time the rule level is raised, and reusing the function values calculated on previous levels.
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The classic method of Gaussian quadrature evaluates the integrand at N+1 points and is constructed to exactly integrate polynomials up to degree 2N+1. In contrast, Clenshaw–Curtis quadrature, above, evaluates the integrand at N+1 points and exactly integrates polynomials only up to degree N. It may seem, therefore, that Clenshaw–Curtis is intrinsically worse than Gaussian quadrature, but in reality this does not seem to be the case. In practice, several authors have observed that Clenshaw–Curtis can have accuracy comparable to that of Gaussian quadrature for the same number of points. This is possible because most numeric integrands are not polynomials (especially since polynomials can be integrated analytically), and approximation of many functions in terms of Chebyshev polynomials converges rapidly (see Chebyshev approximation).
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A first-order extension of the isotropic Gaussian scale space is provided by the affine (Gaussian) scale space. One motivation for this extension originates from the common need for computing image descriptors subject for real-world objects that are viewed under a perspective camera model. To handle such non-linear deformations locally, partial invariance (or more correctly covariance) to local affine deformations can be achieved by considering affine Gaussian kernels with their shapes determined by the local image structure, see the article on affine shape adaptation for theory and algorithms. Indeed, this affine scale space can also be expressed from a non- isotropic extension of the linear (isotropic) diffusion equation, while still being within the class of linear partial differential equations.
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The discovery of algebraic invariants with Gaussian processes is based on David Hilbert's „Über die vollen Invariantensysteme“ and the studies of Grace and Young from 1903. Algebraic invariants with Gaussian processes were discovered by Clemens Par in 2010 in conjunction with the depicted vertical plane. S5 'Signal Analysis' is not specified. It may be either based on statistical methods, which require extensive computational power, or on the discovery of algebraic invariants with Gaussian processes by Clemens Par in 2010 (after having been averted to this classical problem by Rudolf E. Kálmán), based on German mathematician David Hilbert's published proof of the invariant field in 1893 and the apolarity behavior of algebraic cones as extensively studied by Grace and Young in 1903.
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Consider a randomly placed vector r such that both ends of the vector are in the particle. If the vector were held constant in space, while the particle were translated and rotated to any position meeting this condition and an average of the structures were taken, any object would result in a Gaussian mass distribution that would display a Gaussian correlation function, and would appear as an average cloud with no surface. The Fourier transform of results in .
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The Vitali covering theorem is not valid in infinite-dimensional settings. The first result in this direction was given by David Preiss in 1979:. there exists a Gaussian measure γ on an (infinite-dimensional) separable Hilbert space H so that the Vitali covering theorem fails for (H, Borel(H), γ). This result was strengthened in 2003 by Jaroslav Tišer: the Vitali covering theorem in fact fails for every infinite-dimensional Gaussian measure on any (infinite- dimensional) separable Hilbert space..
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When water retention curves are fitted with non-linear least squares, structural overestimation or underestimation can occur. In these cases, the representation of water retention curves can be improved in terms of accuracy and uncertainty by applying Gaussian Process regression to the residuals that are obtained after non-linear least-squares. This is mostly due to the correlation between the datapoints, which is accounted for with Gaussian Process regression through the kernel function. Yousef, B. (June, 2019).
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Most importantly, : ΦG(x) = Φ(x) for x ≥ . Note that Φ() = 0.958…, thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of ΦG(x) is 4/5 = 1.385… > Φ−1(0.9) = 1.282… The following critical values are important in applications: 0.95 = Φ(1.645) = ΦG(1.651), and 0.9 = Φ(1.282) = ΦG(1.386).
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Mandelbrot saw financial markets as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations. In his early work, he found that the price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution.
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The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature. By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space. Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved.
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Because of this relationship, processing time cannot be saved by simulating a Gaussian blur with successive, smaller blurs — the time required will be at least as great as performing the single large blur. Two downscaled images of the Flag of the Commonwealth of Nations. Before downscaling, a Gaussian blur was applied to the bottom image but not to the top image. The blur makes the image less sharp, but prevents the formation of moiré pattern aliasing artifacts.
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In mathematics -- specifically, in the fields of probability theory and inverse problems -- Besov measures and associated Besov-distributed random variables are generalisations of the notions of Gaussian measures and random variables, Laplace distributions, and other classical distributions. They are particularly useful in the study of inverse problems on function spaces for which a Gaussian Bayesian prior is an inappropriate model. The construction of a Besov measure is similar to the construction of a Besov space, hence the nomenclature.
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Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the Deviance information criterion (DIC) -- if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values. see especially Figure 14 and appendix.
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However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the , this window naturally offers a "flat top" to control the amplitude attenuation of a time-series (on which we don't have a control with Gaussian window). In essence, it offers a good (controllable) compromise, in terms of spectral leakage, frequency resolution and amplitude attenuation, between the Gaussian window and the rectangular window. See also for a study on time-frequency representation of this window (or function).
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When such a beam is refocused by a lens, the transverse phase dependence is altered; this results in a different Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so- called waist . At any position relative to the waist (focus) along a beam having a specified , the field amplitudes and phases are thereby determinedSvelto, pp. 153–5. as detailed below.
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A positively chirped ultrashort pulse of light in the time domain. There is no standard definition of ultrashort pulse. Usually the attribute 'ultrashort' applies to pulses with a temporal duration of a few tens of femtoseconds, but in a larger sense any pulse which lasts less than a few picoseconds can be considered ultrafast. A common example is a chirped Gaussian pulse, a wave whose field amplitude follows a Gaussian envelope and whose instantaneous phase has a frequency sweep.
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The observed values in a point process might be modelled as a Poisson process in which the rate (the relevant underlying parameter) is treated as being the exponential of a Gaussian process.
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In case the kernel should also be inferred nonparametrically from the data, the critical filter can be used. Smoothing splines have an interpretation as the posterior mode of a Gaussian process regression.
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If one considers convolution with the kernel instead of with a Gaussian, one obtains the Poisson transform which smoothes and averages a given function in a manner similar to the Weierstrass transform.
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In Gaussian units, the speed of light c appears explicitly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it appears only via the product \mu_0 \varepsilon_0=1/c^2.
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Michel Rolle (21 April 1652 – 8 November 1719) was a French mathematician. He is best known for Rolle's theorem (1691). He is also the co-inventor in Europe of Gaussian elimination (1690).
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Gaussian (software) claims to support COSMO-RS via an external program. SCM licenses a commercial COSMO-RS implementation in the Amsterdam Modeling Suite, which also includes COSMO-SAC, UNIFAC and QSPR models.
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A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision.
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This holds true for any given optical system, and thus the minimum (focussed) spot size or beam waist of a multi-mode laser beam is M times the embedded Gaussian beam waist.
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In quantitative finance, non-Gaussian return distributions are common. The Rachev ratio, as a risk-adjusted performance measurement, characterizes the skewness and kurtosis of the return distribution (see picture on the right).
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The generalized chi-square distribution is obtained from the quadratic form z′Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.
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This proof builds on Lagrange's result that if p=4n+1 is a prime number, then there must be an integer m such that m^2 + 1 is divisible by p (we can also see this by Euler's criterion); it also uses the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since does not divide either of the Gaussian integers m + i and m-i (as it does not divide their imaginary parts), but it does divide their product m^2 + 1, it follows that p cannot be a prime element in the Gaussian integers. We must therefore have a nontrivial factorization of p in the Gaussian integers, which in view of the norm can have only two factors (since the norm is multiplicative, and p^2 = N(p), there can only be up to two factors of p), so it must be of the form p = (x+yi)(x-yi) for some integers x and y. This immediately yields that p = x^2 + y^2.
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In 1946, Dennis Gabor suggested that a signal can be represented in two dimensions, with time and frequency coordinates. And the signal can be expanded into a discrete set of Gaussian elementary signals.
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A system of equations Ax = b for b\in \R^n can be solved by an efficient form of Gaussian elimination when A is tridiagonal called tridiagonal matrix algorithm, requiring O(n) operations.
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In this way, we can define affine-adapted versions of the Laplacian/Difference of Gaussian operator, the determinant of the Hessian and the Hessian-Laplace operator (see also Harris-Affine and Hessian-Affine).
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Christoffel generalized the Gaussian quadrature method for integration and, in connection to this, he also introduced the Christoffel–Darboux formula for Legendre polynomials (he later also published the formula for general orthogonal polynomials).
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A lot of very common probability distributions belong to the class of EDMs, among them are: normal distribution, Binomial distribution, Poisson distribution, Negative binomial distribution, Gamma distribution, Inverse Gaussian distribution, and Tweedie distribution.
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This definition of Euclidean division may be interpreted geometrically in the complex plane (see the figure), by remarking that the distance from a complex number to the closest Gaussian integer is at most .
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There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the CGS system. These include the Gaussian units and the Heaviside–Lorentz units.
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Body essence is an entity invariant to interface reflection, and has two degrees of freedom. The Gaussian coefficient generalizes a conventional simple thresholding scheme, and it provides detailed use of body color similarity.
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This identify is useful in developing a Bayes estimator for multivariate Gaussian distributions. The identity also finds applications in random matrix theory by relating determinants of large matrices to determinants of smaller ones.
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Neural Tangents is a free and open-source Python library used for computing and doing inference with the infinite width NTK and Neural network Gaussian process (NNGP) corresponding to various common ANN architectures.
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The inability to achieve the required lateral navigation accuracy may be due to navigation errors related to aircraft tracking and positioning. The three main errors are path definition error (PDE), flight technical error (FTE) and navigation system error (NSE). The distribution of these errors is assumed to be independent, zero-mean and Gaussian. Therefore, the distribution of total system error (TSE) is also Gaussian with a standard deviation equal to the root sum square (RSS) of the standard deviations of these three errors.
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Masreliez theorem describes a recursive algorithm within the technology of extended Kalman filter, named after the Swedish-American physicist John Masreliez, who is its author. The algorithm estimates the state of a dynamic system with the help of often incomplete measurements marred by distortion.T. Cipra & A. Rubio; Kalman filter with a non-linear non-Gaussian observation relation, Springer (1991). Masreliez's theorem produces estimates that are quite good approximations to the exact conditional mean in non-Gaussian additive outlier (AO) situations.
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Gaussian Quantum Monte Carlo is a quantum Monte Carlo method that shows a potential solution to the fermion sign problem without the deficiencies of alternative approaches. Instead of the Hilbert space, this method works in the space of density matrices that can be spanned by an over-complete basis of gaussian operators using only positive coefficients. Containing only quadratic forms of the fermionic operators, no anti-commuting variables occur and any quantum state can be expressed as a real probability distribution.
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Buckminsterfullerene, C60 In 1970, John Pople developed the Gaussian program greatly easing computational chemistry calculations.W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Gaussian 70 (Quantum Chemistry Program Exchange, Program No. 237, 1970). In 1971, Yves Chauvin offered an explanation of the reaction mechanism of olefin metathesis reactions. In 1975, Karl Barry Sharpless and his group discovered stereoselective oxidation reactions including Sharpless epoxidation,Hill, J. G.; Sharpless, K. B.; Exon, C. M.; Regenye, R. Org. Synth.
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The FT is, however, more general than the Green–Kubo Relations because, unlike them, the FT applies to fluctuations far from equilibrium. In spite of this fact, no one has yet been able to derive the equations for nonlinear response theory from the FT. The FT does not imply or require that the distribution of time-averaged dissipation is Gaussian. There are many examples known when the distribution is non-Gaussian and yet the FT still correctly describes the probability ratios.
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Johnson–Nyquist noise (more often thermal noise) is unavoidable, and generated by the random thermal motion of charge carriers (usually electrons), inside an electrical conductor, which happens regardless of any applied voltage. Thermal noise is approximately white, meaning that its power spectral density is nearly equal throughout the frequency spectrum. The amplitude of the signal has very nearly a Gaussian probability density function. A communication system affected by thermal noise is often modelled as an additive white Gaussian noise (AWGN) channel.
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It is therefore sometimes said that the expansion is bi-orthogonal since the random coefficients are orthogonal in the probability space while the deterministic functions are orthogonal in the time domain. The general case of a process that is not centered can be brought back to the case of a centered process by considering which is a centered process. Moreover, if the process is Gaussian, then the random variables are Gaussian and stochastically independent. This result generalizes the Karhunen–Loève transform.
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In the second approach, terahertz images are developed based on the time delay of the received pulse. In this approach, thicker parts of the objects are well recognized as the thicker parts cause more time delay of the pulse. Energy of the laser spots are distributed by a Gaussian function. The geometry and behavior of Gaussian beam in the Fraunhofer region imply that the electromagnetic beams diverge more as the frequencies of the beams decrease and thus the resolution decreases.
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In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.
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In practical applications, Gaussian process models are often evaluated on a grid leading to multivariate normal distributions. Using these models for prediction or parameter estimation using maximum likelihood requires evaluating a multivariate Gaussian density, which involves calculating the determinant and the inverse of the covariance matrix. Both of these operations have cubic computational complexity which means that even for grids of modest sizes, both operations can have a prohibitive computational cost. This drawback led to the development of multiple approximation methods.
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In statistics, a Tsallis distribution is a probability distribution derived from the maximization of the Tsallis entropy under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy.Tsallis, C. (2009) "Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years", Braz.
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Because the product of two GTOs can be written as a linear combination of GTOs, integrals with Gaussian basis functions can be written in closed form, which leads to huge computational savings (see John Pople). Dozens of Gaussian-type orbital basis sets have been published in the literature. Basis sets typically come in hierarchies of increasing size, giving a controlled way to obtain more accurate solutions, however at a higher cost. The smallest basis sets are called minimal basis sets.
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Unguided electromagnetic waves in free space, or in a bulk isotropic dielectric, can be described as a superposition of plane waves; these can be described as TEM modes as defined below. However in any sort of waveguide where boundary conditions are imposed by a physical structure, a wave of a particular frequency can be described in terms of a transverse mode (or superposition of such modes). These modes generally follow different propagation constants. When two or more modes have an identical propagation constant along the waveguide, then there is more than one modal decomposition possible in order to describe a wave with that propagation constant (for instance, a non-central Gaussian laser mode can be equivalently described as a superposition of Hermite-Gaussian modes or Laguerre-Gaussian modes which are described below).
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Gaussian curvature is 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. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature. Formally, Gaussian curvature only depends on the Riemannian metric of the surface.
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Meanwhile, scientists developed a yet another fully coherent absolute system, which came to be called the Gaussian system, in which the units for purely electrical quantities are taken from CGE-ESU, while the units for magnetic quantities are taken from the CGS- EMU. This system proved very convenient for scientific work and is still widely used. However, the sizes of its units remained either too large or too small—by many orders of magnitude—for practical applications. Finally, on top of all this, in both CGS-ESU and CGS-EMU as well as in the Gaussian system, Maxwell's equations are ‘unrationalized', meaning that they contain various factors of that many workers found awkward. So yet another system was developed to rectify that: the ‘rationalized’ Gaussian system, usually called the Lorentz–Heaviside system.
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In information theory and statistics, negentropy is used as a measure of distance to normality.Aapo Hyvärinen, Survey on Independent Component Analysis, node32: Negentropy, Helsinki University of Technology Laboratory of Computer and Information ScienceAapo Hyvärinen and Erkki Oja, Independent Component Analysis: A Tutorial, node14: Negentropy, Helsinki University of Technology Laboratory of Computer and Information ScienceRuye Wang, Independent Component Analysis, node4: Measures of Non-Gaussianity Out of all distributions with a given mean and variance, the normal or Gaussian distribution is the one with the highest entropy. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes if and only if the signal is Gaussian.
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Common examples of symmetries which lend themselves to Gauss's law include: cylindrical symmetry, planar symmetry, and spherical symmetry. See the article Gaussian surface for examples where these symmetries are exploited to compute electric fields.
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Hence, one may say that the primary way to generate a scale space is by the diffusion equation, and that the Gaussian kernel arises as the Green's function of this specific partial differential equation.
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Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss–Kronrod quadrature rules can be useful.
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See also violet noise, which is a 6 dB increase per octave. Strictly, Brownian motion has a Gaussian probability distribution, but "red noise" could apply to any signal with the 1/f 2 frequency spectrum.
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The Hessian matrix is commonly used for expressing image processing operators in image processing and computer vision (see the Laplacian of Gaussian (LoG) blob detector, the determinant of Hessian (DoH) blob detector and scale space).
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In power engineering, Kron reduction is a method used to reduce or eliminate the desired node without need of repeating the steps like in Gaussian elimination. It is named after American electrical engineer Gabriel Kron.
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If we make a vector of the values of f at N points, x1, ..., xN, in the D-dimensional space, then the vector (f(x1), ..., f(xN)) will always be distributed as a multivariate Gaussian.
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If the channel matrix is completely known, singular value decomposition (SVD) precoding is known to achieve the MIMO channel capacity.E. Telatar, Capacity of multiantenna Gaussian channels , European Transactions on Telecommunications, vol. 10, no. 6, pp.
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When the mean is not known, the minimum mean squared error estimate of the variance of a sample from Gaussian distribution is achieved by dividing by n + 1, rather than n − 1 or n + 2\.
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Accordingly, equation 4 describes the correlogram as shown in Figure 4. One can see that the distribution of the intensity is formed by a Gaussian envelope and a periodic modulation with the period \lambda_0/2.
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Mode (position of apex, most probable value) is calculated using derivative of formula 2; inverse of scaled complementary error function erfcxinv() is used for calculation. The apex is always located on the original (unmodified) Gaussian.
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3 The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject., p. 789 Some authors use the term Gaussian elimination to refer only to the procedure until the matrix is in echelon form, and use the term Gauss–Jordan elimination to refer to the procedure which ends in reduced echelon form. The name is used because it is a variation of Gaussian elimination as described by Wilhelm Jordan in 1888.
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If, for example, the leading coefficient of one of the rows is very close to zero, then to row-reduce the matrix, one would need to divide by that number. This means that any error existed for the number that was close to zero would be amplified. Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable, when using partial pivoting, even though there are examples of stable matrices for which it is unstable.
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For diatomic molecules, a systematic study using a minimum basis set and the first calculation with a larger basis set were published by Ransil and Nesbet respectively in 1960. The first polyatomic calculations using Gaussian orbitals were performed in the late 1950s. The first configuration interaction calculations were performed in Cambridge on the EDSAC computer in the 1950s using Gaussian orbitals by Boys and coworkers. By 1971, when a bibliography of ab initio calculations was published, the largest molecules included were naphthalene and azulene.
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In differential geometry, a smooth surface in three dimensions has a parabolic point when the Gaussian curvature is zero. Typically such points lie on a curve called the parabolic line which separates the surface into regions of positive and negative Gaussian curvature. Points on the parabolic line give rise to folds on the Gauss map: where a ridge crosses a parabolic line there is a cusp of the Gauss map.Ian R. Porteous (2001) Geometric Differentiation, Chapter 11 Ridges and Ribs, pp 182-97, Cambridge University Press .
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Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a linear combination of atomic orbitals. These atomic orbitals are called Slater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more Gaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time. Various basis sets are used in practice, most of which are composed of Gaussian functions.
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A two dimensional convolution matrix is precomputed from the formula and convolved with two dimensional data. Each element in the resultant matrix new value is set to a weighted average of that elements neighborhood. The focal element receives the heaviest weight (having the highest Gaussian value) and neighboring elements receive smaller weights as their distance to the focal element increases. In Image processing, each element in the matrix represents a pixel attribute such as brightness or a color intensity, and the overall effect is called Gaussian blur.
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We have to solve this system for to construct the interpolant p(x). The matrix on the left is commonly referred to as a Vandermonde matrix. The condition number of the Vandermonde matrix may be large, causing large errors when computing the coefficients if the system of equations is solved using Gaussian elimination. Several authors have therefore proposed algorithms which exploit the structure of the Vandermonde matrix to compute numerically stable solutions in O(n2) operations instead of the O(n3) required by Gaussian elimination.
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Roughly speaking this lemma states that geodesics starting at the base point must cut the spheres of fixed radius centred on the base point at right angles. Geodesic polar coordinates are obtained by combining the exponential map with polar coordinates on tangent vectors at the base point. The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated Theorema Egregium.
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Carl Friedrich Gauss introduced his constant to the world in his 1809 Theoria Motus. Piazzi's discovery of Ceres, described in his book Della scoperta del nuovo pianeta Cerere Ferdinandea, demonstrated the utility of the Gaussian gravitation constant in predicting the positions of objects within the Solar System. The Gaussian gravitational constant (symbol ) is a parameter used in the orbital mechanics of the solar system. It relates the orbital period to the orbit's semi-major axis and the mass of the orbiting body in Solar masses.
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Living Electronic Music, p.48. . Each instrument is conceived as a molecule obeying the Maxwell–Boltzmann distribution law,Randel, Don Michael (1996). The Harvard Biographical Dictionary of Music, p.999. . with Gaussian distribution of temperature fluctuation.
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There is an explicit representation for stationary Gaussian processes. A simple example of this representation is : X_t = \cos(at) \xi_1 + \sin(at) \xi_2 where \xi_1 and \xi_2 are independent random variables with the standard normal distribution.
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The original application of chordal completion described in Computers and Intractability involves Gaussian elimination for sparse matrices. During the process of Gaussian elimination, one wishes to minimize fill-in, coefficients of the matrix that were initially zero but later become nonzero, because the need to calculate the values of these coefficients slows down the algorithm. The pattern of nonzeros in a sparse symmetric matrix can be described by an undirected graph (having the matrix as its adjacency matrix); the pattern of nonzeros in the filled-in matrix is always a chordal graph, any minimal chordal completion corresponds to a fill-in pattern in this way. If a chordal completion of a graph is given, a sequence of steps in which to perform Gaussian elimination to achieve this fill-in pattern can be found by computing an elimination ordering of the resulting chordal graph.
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EURANDOM He is author of the book "Large deviations for Gaussian queues", and is associate editor of the journals Stochastic Models and Queuing Systems. He contributed to the book Queues and Lévy fluctuation theory, published in 2015.
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One of the simplest forms of the Langevin equation is when its "noise term" is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection–diffusion equation. However, the Langevin equation is more general.
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Georgios Sivilioglou, et al. successfully fabricated an Airy beam in 2007. A beam with a Gaussian distribution was modulated by a spatial light modulator to have an Airy distribution. The result was recorded by a CCD camera.
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3, Montevideo, Uruguay, p. 227–257. coal mining,C. Ö. Karacan, Ricardo A. Olea (2013), Sequential Gaussian co-simulation of rate decline parameters of longwall gob gas ventholes: International Journal of Rock Mechanics and Mining Sciences, v.
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In the mathematical theory of probability, the Heyde theorem is the characterization theorem concerning the normal distribution (the Gaussian distribution) by the symmetry of one linear form given another. This theorem was proved by C. C. Heyde.
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Older systems used Gaussian-shaped beams and scanned these beams in a raster fashion. Newer systems use shaped beams, which may be deflected to various positions in the writing field (this is also known as vector scan).
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Gaussian 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.A. Lipson, S.G. Lipson, H. Lipson, Optical Physics, 4th edition, 2010, University Press, Cambridge, UK, p. 51. 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.
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In applied mathematics, a steerable filter is an orientation-selective convolution kernel used for image enhancement and feature extraction that can be expressed via a linear combination of a small set of rotated versions of itself. As an example, the oriented first derivative of a 2D Gaussian is a steerable filter. The oriented first order derivative can be obtained by taking the dot product of a unit vector oriented in a specific direction with the gradient. The basis filters are the partial derivatives of a 2D Gaussian with respect to x and y.
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Gauss on his deathbed (1855) Albani Cemetery in Göttingen, Germany Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula.
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Tsallis conjectured in 1999 (Brazilian Journal of Physics 29, 1; Figure 4): #That a longstanding quasi-stationary state (QSS) was expected in long-range interacting Hamiltonian systems (one of the core problems of statistical mechanics). This was verified by groups around the world. #That this QSS should be described by Tsallis statistics instead of Boltzmann–Gibbs statistics. This was verified in June 2007 by Pluchino, Rapisarda and Tsallis (in the last figure, instead of the Maxwellian (Gaussian) distribution of velocities (valid for short-range interactions), one sees a q-Gaussian).
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The Feature- based Morphometry (FBM) technique uses extrema in a difference of Gaussian scale-space to analyze and classify 3D magnetic resonance images (MRIs) of the human brain. FBM models the image probabilistically as a collage of independent features, conditional on image geometry and group labels, e.g. healthy subjects and subjects with Alzheimer's disease (AD). Features are first extracted in individual images from a 4D difference of Gaussian scale- space, then modeled in terms of their appearance, geometry and group co- occurrence statistics across a set of images.
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Thus it appears unlikely, but not impossible, that the cold spot was generated by the standard mechanism of quantum fluctuations during cosmological inflation, which in most inflationary models gives rise to Gaussian statistics. The cold spot may also, as suggested in the references above, be a signal of non-Gaussian primordial fluctuations. Some authors called into question the statistical significance of this cold spot. In 2013, the CMB Cold Spot was also observed by the Planck satellite at similar significance, discarding the possibility of being caused by a systematic error of the WMAP satellite.
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Without ε0, the two sides would not have consistent dimensions in SI, whereas the quantity ε0 does not appear in Gaussian equations. This is an example of how some dimensional physical constants can be eliminated from the expressions of physical law simply by the judicious choice of units. In SI, 1/ε0, converts or scales flux density, D, to electric field, E (the latter has dimension of force per charge), while in rationalized Gaussian units, electric flux density is the same quantity as electric field strength in free space.
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There exists a more general extension of the Gaussian scale-space model to affine and spatio-temporal scale-spaces.Lindeberg, T. Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio- temporal scale-space, Journal of Mathematical Imaging and Vision, 40(1): 36–81, 2011.Lindeberg, T. Generalized axiomatic scale-space theory, Advances in Imaging and Electron Physics, Elsevier, volume 178, pages 1–96, 2013.T. Lindeberg (2016) "Time-causal and time-recursive spatio-temporal receptive fields", Journal of Mathematical Imaging and Vision, 55(1): 50–88.
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Spacetime wave packets are a form of spatially correlated light that seems to violate the normal physical rules applying to light beams. In particular, their group velocity in free space can differ from the normal speed of light in vacuum. and their behavior under refraction does not follow the normal expectations given by Snell's law. Monochromatic Gaussian beam is shown to be transformed into spacetime wave packets under Lorentz transformation, thus any monochromatic Gaussian beam observed in a reference frame moving at relativistic velocity appears as spacetime wave packets.
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Diagram of surface reflection The surface roughness model used in the derivation of the Oren-Nayar model is the microfacet model, proposed by Torrance and Sparrow, which assumes the surface to be composed of long symmetric V-cavities. Each cavity consists of two planar facets. The roughness of the surface is specified using a probability function for the distribution of facet slopes. In particular, the Gaussian distribution is often used, and thus the variance of the Gaussian distribution, \sigma^2, is a measure of the roughness of the surfaces.
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Because every child must be conjugate to its parent, this limits the types of distributions that can be used in the model. For example, the parents of a Gaussian distribution must be a Gaussian distribution (corresponding to the Mean) and a gamma distribution (corresponding to the precision, or one over \sigma in more common parameterizations). Discrete variables can have Dirichlet parents, and Poisson and exponential nodes must have gamma parents. However, if the data can be modeled in this manner, VMP offers a generalized framework for providing inference.
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Modeling the changes by distributions with finite variance is now known to be inappropriate. Benoît Mandelbrot found in the 1960s that changes in prices in financial markets do not follow a Gaussian distribution, but are rather modeled better by Lévy stable distributions. The scale of change, or volatility, depends on the length of the time interval to a power a bit more than 1/2. Large changes up or down, also called fat tails, are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation.
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Each pixel's new value is set to a weighted average of that pixel's neighborhood. The original pixel's value receives the heaviest weight (having the highest Gaussian value) and neighboring pixels receive smaller weights as their distance to the original pixel increases. This results in a blur that preserves boundaries and edges better than other, more uniform blurring filters; see also scale space implementation. In theory, the Gaussian function at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel.
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In the case of equations in unknowns, it requires computation of determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. Cramer's rule can also be numerically unstable even for 2×2 systems. However, it has recently been shown that Cramer's rule can be implemented in O(n3) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination (consistently requiring 2.5 times as many arithmetic operations for all matrix sizes), while exhibiting comparable numeric stability in most cases.
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In the procedure of hypothesis testing, one needs to form the joint distribution of test statistics to conduct the test and control type I errors. However, the true distribution is often unknown and a proper null distribution ought to be used to represent the data. For example, one sample and two samples tests of means can use t statistics which have Gaussian null distribution, while F statistics, testing k groups of population means, which have Gaussian quadratic form the null distribution.Dudoit, S., and M. J. Van Der Laan.
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For instance, better Euclidean solutions can be found using k-medians and k-medoids. The problem is computationally difficult (NP-hard); however, efficient heuristic algorithms converge quickly to a local optimum. These are usually similar to the expectation-maximization algorithm for mixtures of Gaussian distributions via an iterative refinement approach employed by both k-means and Gaussian mixture modeling. They both use cluster centers to model the data; however, k-means clustering tends to find clusters of comparable spatial extent, while the expectation-maximization mechanism allows clusters to have different shapes.
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The TRAQSIM model was developed in 2004 as part of a Ph.D dissertation with support by the U.S. Department of Transportation's Volpe National Transportation Systems Center Air Quality Facility. The model incorporates dynamic vehicle behavior with a non-steady state Gaussian puff algorithm. Unlike HYROAD, TRAQSIM combines traffic simulation, second-by-second modal emissions, and Gaussian puff dispersion into a fully integrated system (a true simulation) that models individual vehicles as discrete moving sources. TRAQSIM was developed as a next generation model to be the successor to the current CALINE3 and CAL3QHC regulatory models.
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Some quantum chemistry software uses sets of Slater-type functions (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which give a displaced Gaussian) has led many to expand them in terms of Gaussians. Analytical ab initio software for polyatomic molecules has been developed, e.g., STOP: a Slater Type Orbital Package in 1996.
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In laser science, the beam parameter product (BPP) is the product of a laser beam's divergence angle (half-angle) and the radius of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it can be focused to a small spot. A Gaussian beam has the lowest possible BPP, \lambda/\pi, where \lambda is the wavelength of the light. The ratio of the BPP of an actual beam to that of an ideal Gaussian beam at the same wavelength is denoted M2 ("M squared").
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Rachev's academic work on non-Gaussian models in mathematical finance was inspired by the difficulties of common classical Gaussian-based models to capture empirical properties of financial data. Rachev and his daughter, Borjana Racheva-Iotova, established Bravo Group in 1999, a company with the goal to develop software based on Rachev's research on fat-tailed models. The company was later acquired by FinAnalytica. The company has won the Waters Rankings "Best Market Risk Solution Provider" award in 2010, 2012, and 2015, and also the "Most Innovative Specialist Vendor" Risk Award in 2014.
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Through the 1930s, progressively more general proofs of the Central Limit Theorem were presented. Many natural systems were found to exhibit Gaussian distributions—a typical example being height distributions for humans. When statistical methods such as analysis of variance became established in the early 1900s, it became increasingly common to assume underlying Gaussian distributions. A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's College at the University of Cambridge.
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Gaussian beam width as a function of the distance along the beam, which forms a hyperbola. : beam waist; : depth of focus; : Rayleigh range; : total angular spread The shape of a Gaussian beam of a given wavelength is governed solely by one parameter, the beam waist . This is a measure of the beam size at the point of its focus ( in the above equations) where the beam width (as defined above) is the smallest (and likewise where the intensity on-axis () is the largest). From this parameter the other parameters describing the beam geometry are determined.
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Xavier Fernique Xavier Fernique (3 May 1934 – 15 March 2020) was a mathematician, noted mostly for his contributions to the theory of stochastic processes. Fernique's theorem, a result on the integrability of Gaussian measures, is named after him.
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J. Møller, A. R. Syversveen, and R. P. Waagepetersen. Log Gaussian Cox Processes. Scandinavian journal of statistics, 25(3):451–482, 1998. More generally, the intensity measures is a realization of a non-negative locally finite random measure.
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Computing the rank of a tensor of order greater than 2 is NP-hard. Therefore, if , there cannot be a polynomial time analog of Gaussian elimination for higher-order tensors (matrices are array representations of order-2 tensors).
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The Unsharp Mask tool is considered to give more targeted results for photographs than a normal sharpening filter. The Selective Gaussian Blur tool works in a similar way, except it blurs areas of an image with little detail.
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Academic Press, 2008, p. 88. where x is the distance from the origin in the horizontal axis, y is the distance from the origin in the vertical axis, and σ is the standard deviation of the Gaussian distribution.
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While the Anscombe transform is appropriate for pure Poisson data, in many applications the data presents also an additive Gaussian component. These cases are treated by a Generalized Anscombe transform and its asymptotically unbiased or exact unbiased inverses.
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Comparison of Gaussian (red) and Lorentzian (blue) standardized line shapes. The HWHM (w/2) is 1. Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6.
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In the past, Gaussian, Inc. has attracted controversy for its licensing terms that stipulate that researchers who develop competing software packages are not permitted to use the software. Some scientists consider these terms overly restrictive. The anonymous group bannedbygaussian.
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Therefore, it has a unique solution, and (with the outer face fixed) the graph has a unique Tutte embedding. This embedding can be found in polynomial time by solving the system of equations, for instance by using Gaussian elimination.
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Figure 7. Energy diagram illustrating the Franck–Condon principle applied to the solvation of chromophores. The parabolic potential curves symbolize the interaction energy between the chromophores and the solvent. The Gaussian curves represent the distribution of this interaction energy.
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A long- standing open problem, posed by Mikhail Gromov, concerns the calculation of the filling area of the Riemannian circle. The filling area is conjectured to be 2, a value attained by the hemisphere of constant Gaussian curvature +1.
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"Facial Emotion Detection Considering Partial Occlusion Of Face Using Baysian Network". Computers and Informatics (2011): 96–101. , Gaussian Mixture modelsHari Krishna Vydana, P. Phani Kumar, K. Sri Rama Krishna and Anil Kumar Vuppala. "Improved emotion recognition using GMM-UBMs".
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To adjust for beam divergence a second car on the linear stage with two lenses can be used. The two lenses act as a telescope producing a flat phase front of a Gaussian beam on a virtual end mirror.
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In a similar procedure to how the normal distribution can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy, the q-Gaussian can be derived from a maximization of the Tsallis entropy subject to the appropriate constraints.
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These Gaussian derivative operators can in turn be combined by linear or non-linear operators into a larger variety of different types of feature detectors, which in many cases can be well modelled by differential geometry. Specifically, invariance (or more appropriately covariance) to local geometric transformations, such as rotations or local affine transformations, can be obtained by considering differential invariants under the appropriate class of transformations or alternatively by normalizing the Gaussian derivative operators to a locally determined coordinate frame determined from e.g. a preferred orientation in the image domain or by applying a preferred local affine transformation to a local image patch (see the article on affine shape adaptation for further details). When Gaussian derivative operators and differential invariants are used in this way as basic feature detectors at multiple scales, the uncommitted first stages of visual processing are often referred to as a visual front-end.
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This criterion, firstly described by and now adopted in propagation codes like, allows one to determine the realm of application of near and far field approximations taking into account the actual wavefront surface shape at the observation point, to sample its phase without aliasing. This criterion is named Gaussian pilot beam and fixes the best propagation method (among angular spectrum, Fresnel and Fraunhofer diffraction) by looking at the behavior of a Gaussian beam piloted from the aperture position and the observation position. Near/far field approximations are fixed by the analytical calculation of the Gaussian beam Rayleigh length and by its comparison with the input/output propagation distance. If the ratio between input/output propagation distance and Rayleigh length returns \le 1 the surface wavefront maintains itself nearly flat along its path, which means that no sampling rescaling is requested for the phase measurement.
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When two or more parameters of a fitting curve are not known the method of non-linear least squares must be used.Gans, Section 8.3, Gaussian, Lorentzian and related functions The reliability of curve fitting in this case is dependent on the separation between the components, their shape functions and relative heights, and the signal-to-noise ratio in the data. When Gaussian- shaped curves are used for the decomposition of set of Nsol spectra into Npks curves, the p_0 and w parameters are common to all Nsol spectra. This allows to calculated the heights of each Gaussian curve in each spectrum (Nsol·Npks parameters) by a (fast) linear least squares fitting procedure, while the p_0 and w parameters (2·Npks parameters) can be obtained with a non-linear least- square fitting on the data from all spectra simultaneously, thus reducing dramatically the correlation between optimized parameters.
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In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian process, indexed by this set, satisfying the conditions below.
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This section has a list of the basic formulae of electromagnetism, given in Lorentz–Heaviside, Gaussian and SI units. Most symbol names are not given; for complete explanations and definitions, please click to the appropriate dedicated article for each equation.
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Circular symmetry of complex random variables is a common assumption used in the field of wireless communication. A typical example of a circular symmetric complex random variable is the complex Gaussian random variable with zero mean and zero pseudo-covariance matrix.
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Photons in a hypergeometric-Gaussian beam have an orbital angular momentum of mħ. The integer m also gives the strength of the vortex at the beam's centre. Spin angular momentum of circularly polarized light can be converted into orbital angular momentum.
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Adding controlled noise from predetermined distributions is a way of designing differentially private mechanisms. This technique is useful for designing private mechanisms for real-valued functions on sensitive data. Some commonly used distributions for adding noise include Laplace and Gaussian distributions.
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Ellen Gethner is a US mathematician and computer scientist specializing in graph theory who won the Mathematical Association of America's Chauvenet Prize in 2002 with co-authors Stan Wagon and Brian Wick for their paper A stroll through the Gaussian Primes.
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The Fermi filter is a common image processing filter that uses the Fermi-Dirac distribution in the Frequency domain to perform a low-pass filter or high-pass filter similar to a Gaussian blur, but the harshness can be scaled.
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In particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the Fermi wavevector, i.e. the long-distance limit. This article uses cgs-Gaussian units.
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The Anisotropic Network Model was introduced in 2000 (Atilgan et al., 2001; Doruker et al., 2000), inspired by the pioneering work of Tirion (1996), succeeded by the development of the Gaussian network model (GNM) (Bahar et al., 1997; Haliloglu et al.
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Bakirov, N.K. and Székely, G. J (2005). "Students’ t-test for Gaussian scale mixtures" (alternative link) Zapiski Nauchnyh Seminarov POMI, 328, Probability and Statistics. Part 9 (editor V.N.Sudakov) 5-19\. Reprinted (2006): Journal of Mathematical Sciences, 139 (3) 6497-6505 .
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Since the linear equations require O(n^3) operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large n for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non- singular problems.
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109-193, in Advances in Infrared and Raman Spectroscopy, Volume 4 (1978), Editors Clark, R.J.H; Hester, R.E. Line maxima may also be shifted. Because there are many sources of broadening, the lines have a stable distribution, tending towards a Gaussian shape.
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The kernel density estimates are sums of Gaussians and may therefore be represented as Gaussian mixture models (GMM). Jian and Vemuri use the GMM version of the KC registration algorithm to perform non-rigid registration parametrized by thin plate splines.
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Finally, note that because the variables x and y are jointly Gaussian, the minimum MSE estimator is linear.See the article minimum mean square error. Therefore, in this case, the estimator above minimizes the MSE among all estimators, not only linear estimators.
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The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.
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Control systems have a need for smoothing filters in their feedback loops with criteria to maximise the speed of movement of a mechanical system to the prescribed mark and at the same time minimise overshoot and noise induced motions. A key problem here is the extraction of Gaussian signals from a noisy background. An early paper on this was published during WWII by Norbert Wiener with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman (Kalman filter) later reformulated this in terms of state-space smoothing and prediction where it is known as the linear-quadratic-Gaussian control problem.
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Another instance of the separation principle arises in the setting of linear stochastic systems, namely that state estimation (possibly nonlinear) together with an optimal state feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output measurements. When process and observation noise are Gaussian, the optimal solution separates into a Kalman filter and a linear-quadratic regulator. This is known as linear-quadratic- Gaussian control. More generally, under suitable conditions and when the noise is a martingale (with possible jumps), again a separation principle applies and is known as the separation principle in stochastic control. . . . .
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In statistics and machine learning, Gaussian process approximation is a computational method that accelerates inference tasks in the context of a Gaussian process model, most commonly likelihood evaluation and prediction. Like approximations of other models, they can often be expressed as additional assumptions imposed on the model, which do not correspond to any actual feature, but which retain its key properties while simplifying calculations. Many of these approximation methods can be expressed in purely linear algebraic or functional analytic terms as matrix or function approximations. Others are purely algorithmic and cannot easily be rephrased as a modification of a statistical model.
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One is thus making a distinction between the experimental variogram that is a visualisation of a possible spatial/temporal correlation and the variogram model that is further used to define the weights of the kriging function. Note that the experimental variogram is an empirical estimate of the covariance of a Gaussian process. As such, it may not be positive definite and hence not directly usable in kriging, without constraints or further processing. This explains why only a limited number of variogram models are used: most commonly, the linear, the spherical, the Gaussian and the exponential models.
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In computer vision, the Marr–Hildreth algorithm is a method of detecting edges in digital images, that is, continuous curves where there are strong and rapid variations in image brightness. The Marr–Hildreth edge detection method is simple and operates by convolving the image with the Laplacian of the Gaussian function, or, as a fast approximation by difference of Gaussians. Then, zero crossings are detected in the filtered result to obtain the edges. The Laplacian-of-Gaussian image operator is sometimes also referred to as the Mexican hat wavelet due to its visual shape when turned upside-down.
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Thomas Royen (born July 6, 1947 in Frankfurt am Main) is a retired German professor of statistics who has been affiliated with the University of Applied Sciences Bingen. Royen came to prominence in the spring of 2017 for a relatively simple proof for the Gaussian Correlation Inequality (GCI), a conjecture that originated in the 1950s, which he had published three years earlier without much recognition., Royen's proof of the Gaussian correlation inequality, arXiv:1512.08776 A proof of this conjecture, which lies at the intersection of geometry, probability theory and statistics, had eluded top experts for decades.
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SI units predominate in most fields, and continue to increase in popularity at the expense of Gaussian units. Alternative unit systems also exist. Conversions between quantities in the Gaussian unit system and the SI unit system are not as straightforward as direct unit conversions because the quantities themselves are defined differently in the different systems, which has the effect that the equations expressing physical laws of electromagnetism (such as Maxwell's equations) change depending on what system of units is being used. As an example, quantities that are dimensionless in one system may have dimension in another.
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The M2 parameter is a measure of beam quality; a low M2 value indicates good beam quality and ability to be focused to a tight spot. The value M is equal to the ratio of the beam's angle of divergence to that of a Gaussian beam with the same D4σ waist width. Since the Gaussian beam diverges more slowly than any other beam shape, the M2 parameter is always greater than or equal to one. Other definitions of beam quality have been used in the past, but the one using second moment widths is most commonly accepted.
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Thus, the Gaussian moat problem may be phrased in a different but equivalent form: is there a finite bound on the widths of the moats that have finitely many primes on the side of the origin? Computational searches have shown that the origin is separated from infinity by a moat of width 6.. It is known that, for any positive number k, there exist Gaussian primes whose nearest neighbor is at distance k or larger. In fact, these numbers may be constrained to be on the real axis. For instance, the number 20785207 is surrounded by a moat of width 17.
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Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types. Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.
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The resulting beam has a larger diameter, and hence a lower divergence. Divergence of a laser beam may be reduced below the diffraction of a Gaussian beam or even reversed to convergence if the refractive index of the propagation media increases with the light intensity. This may result in a self-focusing effect. When the wave front of the emitted beam has perturbations, only the transverse coherence length (where the wave front perturbation is less than 1/4 of the wavelength) should be considered as a Gaussian beam diameter when determining the divergence of the laser beam.
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In general, the Euclidean algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments. The Euclidean algorithm developed for two Gaussian integers and is nearly the same as that for ordinary integers, but differs in two respects. As before, the task at each step is to identify a quotient and a remainder such that :r_k = r_{k-2} - q_k r_{k-1}, where , where , and where every remainder is strictly smaller than its predecessor: . The first difference is that the quotients and remainders are themselves Gaussian integers, and thus are complex numbers.
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These are standard results in spherical, hyperbolic and high school trigonometry (see below). Gauss generalised these results to an arbitrary surface by showing that the integral of the Gaussian curvature over the interior of a geodesic triangle is also equal to this angle difference or excess. His formula showed that the Gaussian curvature could be calculated near a point as the limit of area over angle excess for geodesic triangles shrinking to the point. Since any closed surface can be decomposed up into geodesic triangles, the formula could also be used to compute the integral of the curvature over the whole surface.
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Hyperbolic paraboloid A model of an elliptic hyperboloid of one sheet A 300px A saddle surface is a smooth surface containing one or more saddle points. Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid z=x^2-y^2 (which is often referred to as "the saddle surface" or "the standard saddle surface") and the hyperboloid of one sheet. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape. Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature.
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Practical laser resonators may contain more than two mirrors; three- and four-mirror arrangements are common, producing a "folded cavity". Commonly, a pair of curved mirrors form one or more confocal sections, with the rest of the cavity being quasi-collimated and using plane mirrors. The shape of the laser beam depends on the type of resonator: The beam produced by stable, paraxial resonators can be well modeled by a Gaussian beam. In special cases the beam can be described as a single transverse mode and the spatial properties can be well described by the Gaussian beam, itself.
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The starting point for Regge's work is the fact that every four dimensional time orientable Lorentzian manifold admits a triangulation into simplices. Furthermore, the spacetime curvature can be expressed in terms of deficit angles associated with 2-faces where arrangements of 4-simplices meet. These 2-faces play the same role as the vertices where arrangements of triangles meet in a triangulation of a 2-manifold, which is easier to visualize. Here a vertex with a positive angular deficit represents a concentration of positive Gaussian curvature, whereas a vertex with a negative angular deficit represents a concentration of negative Gaussian curvature.
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An article published in Polit Online looked at criticisms of the analysis. The supposition of Gaussian distribution was criticized by sociologist Aleksey Grazhdankin, a Deputy Director of Levada Center (top independent non-governmental polling and sociological research organization in Russia). Grazhdankin cites regional differences and the existence of the so-called "electoral enclaves" in Russia, which vote very differently from the surrounding areas, often because the recent rise of the quality of life in such enclaves is associated with the actions of the authorities. Grazhdankin says he does not believe the graphs with non-Gaussian distributions indicate vote fraud.
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The procedure referred to by the term fangcheng and explained in the eighth chapter of The Nine Chapters, is essentially a procedure to find the solution of systems of n equations in n unknowns and is equivalent to certain similar procedures in modern linear algebra. The earliest recorded fangcheng procedure is similar to what we now call Gaussian elimination. The fangcheng procedure was popular in ancient China and was transmitted to Japan. It is possible that this procedure was transmitted to Europe also and served as precursors of the modern theory of matrices, Gaussian elimination, and determinants.
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For a top-hat beam, the upper integration limits may be bounded by rmax, such that r ≤ rmax − R. Thus, the limited grid coverage in the r direction does not affect the convolution. To convolve reliably for physical quantities at r in response to a top-hat beam, we must ensure that rmax in photon transport methods is large enough that r ≤ rmax − R holds. For a Gaussian beam, no simple upper integration limits exist because it theoretically extends to infinity. At r >> R, a Gaussian beam and a top-hat beam of the same R and S0 have comparable convolution results.
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In general such statistics arrive in the presence of heavy-tailed distributions, and the presence of dragon kings will augment the already oversized impact of extreme events. Despite the importance of extreme events, due to ignorance, misaligned incentives, and cognitive biases, there is often a failure to adequately anticipate them. Technically speaking, this leads to poorly specified models where distributions that are not heavy-tailed enough, and under-appreciate both serial and multivariate dependence of extreme events. Some examples of such failures in risk assessment include the use of Gaussian models in finance (Black–Scholes, the Gaussian copula, LTCM), the use of Gaussian processes and linear wave theory failing to predict the occurrence of rogue waves, the failure of economic models in general to predict the financial crisis of 2007–2008, and the under-appreciation of external events, cascades, and nonlinear effects in probabilistic risk assessment, leading to not anticipating the Fukushima Daiichi nuclear disaster in 2011.
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Scale- Space Theory in Computer Vision, Kluwer Academic Publishers, 1994, (see specifically Chapter 2 for an overview of Gaussian and Laplacian image pyramids and Chapter 3 for theory about generalized binomial kernels and discrete Gaussian kernels)See the article on multi-scale approaches for a very brief theoretical statement Thus, given a two-dimensional image, we may apply the (normalized) binomial filter (1/4, 1/2, 1/4) typically twice or more along each spatial dimension and then subsample the image by a factor of two. This operation may then proceed as many times as desired, leading to a compact and efficient multi-scale representation. If motivated by specific requirements, intermediate scale levels may also be generated where the subsampling stage is sometimes left out, leading to an oversampled or hybrid pyramid. With the increasing computational efficiency of CPUs available today, it is in some situations also feasible to use wider support Gaussian filters as smoothing kernels in the pyramid generation steps.
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In other words, any magnetic field line that enters a given volume must somewhere exit that volume. Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.
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This method for solving systems of linear equations based on determinants was found in 1684 by Leibniz (Cramer published his findings in 1750). Although Gaussian elimination requires O(n^3) arithmetic operations, linear algebra textbooks still teach cofactor expansion before LU factorization.
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Lindeberg ``Scale invariant feature transform, Scholarpedia, 7(5):10491, 2012. for the explicit relation between the difference-of-Gaussian operator and the scale-normalized Laplacian operator. This approach is for instance used in the scale-invariant feature transform (SIFT) algorithm—see Lowe (2004).
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The most important applications of semiclassical gravity are to understand the Hawking radiation of black holes and the generation of random gaussian-distributed perturbations in the theory of cosmic inflation, which is thought to occur at the very beginnings of the big bang.
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Hence an interval predictor model can be seen as a guaranteed bound on quantile regression. Interval predictor models can also be seen as a way to prescribe the support of random predictor models, of which a Gaussian process is a specific case .
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The Dawson function, F(x) = D_+(x), around the origin The Dawson function, D_-(x), around the origin In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function.
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The distribution of the sum of weights is approximately Gaussian, with a peak at and width , so that when is approximately equal to the transition occurs. 223 − 1 is about 4 million, while the width of the distribution is only 5 million.
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Detection of Patterns Below Clutter in Images. Int. Conf. On Integration of Knowledge Intensive Multi-Agent Systems, Cambridge, MA Oct.1-3, 2003. a uniform model for noise, Gaussian blobs for highly-fuzzy, poorly resolved patterns, and parabolic models for ‘smiles’ and ‘frowns’.
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A common goal in Bayesian experimental design is to maximise the expected Kullback–Leibler divergence between the prior and the posterior. When posteriors are approximated to be Gaussian distributions, a design maximising the expected Kullback–Leibler divergence is called Bayes d-optimal.
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If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. The method is named after Werner Romberg (1909–2003), who published the method in 1955.
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Feature Extraction and Image Processing. Academic Press, 2008, p. 88.R.A. Haddad and A.N. Akansu, "A Class of Fast Gaussian Binomial Filters for Speech and Image Processing," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 39, pp 723-727, March 1991.
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It is important to note that choosing other wavelets, levels, and thresholding strategies can result in different types of filtering. In this example, white Gaussian noise was chosen to be removed. Although, with different thresholding, it could just as easily have been amplified.
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U(0,1) + 0.5U(0,1) gives a trapezoidal distribution). Actually the Student-t distribution provides a natural extension of the normal Gaussian distribution for modeling of long tail data. And such generalized Bates distribution is doing so for short tail data (kurtosis < 3).
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This can be done by adding Gaussian blur to the shadow's alpha channel before blending. Inset drop shadows are a type which draws the shadows inside the element. This allows the interface element to appear as if it is sunken into the interface.
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Similar to the modified G2(+) method, CBS-QB3 has been modified by the inclusion of diffuse functions in the geometry optimization step to give CBS-QB3(+). The CBS family of methods is available via keywords in the Gaussian 09 suite of programs.
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However, if the expansion coefficients are obtained as functional principal components, then in some cases (e.g. Gaussian predictor function X), they will be independent in which case backfitting is not needed, and one can use popular smoothing methods for estimating the unknown parameter functions f_j.
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An open-source Python library developed by Xanadu for designing, simulating, and optimizing continuous variable (CV) quantum optical circuits. Three simulators are provided - one in the Fock basis, one using the Gaussian formulation of quantum optics, and one using the TensorFlow machine learning library.
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In 1964 he moved to Carnegie Mellon University in Pittsburgh, Pennsylvania, where he had experienced a sabbatical in 1961 to 1962. In 1993 he moved to Northwestern University in Evanston, Illinois where he was Trustees Professor of Chemistry until his death.John Pople Chronology at Gaussian.
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In more algebraic terms, the period lattice is a real multiple of the Gaussian integers. The constants , , and are given by :e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12. The case , may be handled by a scaling transformation. However, this may involve complex numbers.
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The volume and surface area of a cylinder, cone sphere and torus are calculated using pi. Pi is also used in calculating planetary orbit times, gaussian curves and alternating current. In calculus, there are infinite series that involve pi and pi is used in trigonometry.
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3, pp. 747-66, 2008. In variational filtering, an ensemble of particles diffuse over the free energy landscape in a frame of reference that moves with the expected (generalized) motion of the ensemble. This provides a relatively simple scheme that eschews Gaussian (unimodal) assumptions.
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Had the random variable x also been Gaussian, then the estimator would have been optimal. Notice, that the form of the estimator will remain unchanged, regardless of the apriori distribution of x, so long as the mean and variance of these distributions are the same.
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1987 is an odd number and the 300th prime number. It is the first number of a sexy prime triplet (1987, 1993, 1999). Being of the form 4n + 3, it is a Gaussian prime. It is a lucky number and therefore also a lucky prime.
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This section covers some additional formula for evaluating depth of field; however they are all subject to significant simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography, lens designers would use significantly more complex ones.
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This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics. Chapter 16.
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495--534 In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the linear-quadratic-Gaussian control problem.
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The Bellman pseudospectral method takes advantage of the node accumulation at the initial point to anti-alias the solution and discards the remainder of the nodes. Thus the final distribution of nodes is non-Gaussian and dense while the computational method retains a sparse structure.
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In a Bayesian framework, we use Bayes' Theorem to predict the Kriging mean and covariance conditional on the observations. When using GEK, the observations are usually the results of a number of computer simulations. GEK can be interpreted as a form of Gaussian process regression.
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All quantities are in Gaussian (cgs) units except energy and temperature expressed in eV and ion mass expressed in units of the proton mass \mu = m_i/m_p; Z is charge state; k is Boltzmann's constant; K is wavenumber; \ln\Lambda is the Coulomb logarithm.
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Theoretical bit-error rate curves of encoded QPSK (recursive and non-recursive, soft decision), additive white Gaussian noise channel. Curves are small distinguished due to approximately the same free distances and weights. The free distanceMoon, Todd K. "Error correction coding." Mathematical Methods and Algorithms.
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Polynomial chaos can be utilized in the prediction of non-linear functionals of Gaussian stationary increment processes conditioned on their past realizations Daniel Alpay and Alon Kipnis, Wiener Chaos Approach to Optimal Prediction, Numerical Functional Analysis and Optimization, 36:10, 1286-1306, 2015. DOI: 10.1080/01630563.2015.1065273. Specifically, such prediction is obtained by deriving the chaos expansion of the functional with respect to a special basis for the Gaussian Hilbert space generated by the process that with the property that each basis element is either measurable or independent with respect to the given samples. For example, this approach leads to an easy prediction formula for the Fractional Brownian motion.
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Principal sources of Gaussian noise in digital images arise during acquisition. The sensor has inherent noise due to the level of illumination and its own temperature, and the electronic circuits connected to the sensor inject their own share of electronic circuit noise. A typical model of image noise is Gaussian, additive, independent at each pixel, and independent of the signal intensity, caused primarily by Johnson–Nyquist noise (thermal noise), including that which comes from the reset noise of capacitors ("kTC noise"). Amplifier noise is a major part of the "read noise" of an image sensor, that is, of the constant noise level in dark areas of the image.
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The mode, the so-called doughnut mode, is a special case consisting of a superposition of two modes (), rotated with respect to one another. The overall size of the mode is determined by the Gaussian beam radius , and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation. Higher order modes are relatively larger compared to the mode, and thus the fundamental Gaussian mode of a laser may be selected by placing an appropriately sized aperture in the laser cavity. In many lasers, the symmetry of the optical resonator is restricted by polarizing elements such as Brewster's angle windows.
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A Cox point process, Cox process or doubly stochastic Poisson process is a generalization of the Poisson point process by letting its intensity measure \textstyle \Lambda to be also random and independent of the underlying Poisson process. The process is named after David Cox who introduced it in 1955, though other Poisson processes with random intensities had been independently introduced earlier by Lucien Le Cam and Maurice Quenouille. The intensity measure may be a realization of random variable or a random field. For example, if the logarithm of the intensity measure is a Gaussian random field, then the resulting process is known as a log Gaussian Cox process.
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Lorentz–Heaviside units (or Heaviside–Lorentz units) constitute a system of units (particularly electromagnetic units) within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant and magnetic constant do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Lorentz–Heaviside units may be regarded as normalizing and , while at the same time revising Maxwell's equations to use the speed of light instead. Lorentz–Heaviside units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of appearing explicitly in Maxwell's equations.
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Gaussian 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. 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.
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In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
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The standard fiber optical trap relies on the same principle as the optical trapping, but with the Gaussian laser beam delivered through an optical fiber. If one end of the optical fiber is molded into a lens-like facet, the nearly gaussian beam carried by a single mode standard fiber will be focused at some distance from the fiber tip. The effective Numerical Aperture of such assembly is usually not enough to allow for a full 3D optical trap but only for a 2D trap (optical trapping and manipulation of objects will be possible only when, e.g., they are in contact with a surface ).
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Based on the study of statistics of contourlet coefficients of natural images, the HMT model for the contourlet transform is proposed. The statistics show that the contourlet coefficients are highly non-Gaussian, high interaction dependent on all their eight neighbors and high inter-direction dependent on their cousins. Therefore, the HMT model, that captures the highly non-Gaussian property, is used to get the dependence on neighborhood through the links between the hidden states of the coefficients. This HMT model of contourlet transform coefficients has better results than original contourlet transform and other HMT modeled transforms in denoising and texture retrieval, since it restores edges better visually.
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For small scales, a low-order FIR filter may be a better smoothing filter than a recursive filter. The symmetric 3-kernel , for t ≤ 0.5 smooths to a scale of t using a pair of real zeros at Z < 0, and approaches the discrete Gaussian in the limit of small t. In fact, with infinitesimal t, either this two-zero filter or the two-pole filter with poles at Z = t/2 and Z = 2/t can be used as the infinitesimal generator for the discrete Gaussian kernels described above. The FIR filter's zeros can be combined with the recursive filter's poles to make a general high-quality smoothing filter.
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Isaac Newton himself determined a value of this constant which agreed with Gauss' value to six significant digits."The numerical value of the Gaussian constant was determined by Newton himself 120 years prior to Gauss. It agrees with the modern value to six significant figures. Hence the name 'Gaussian constant' should be regarded as a tribute to Gauss' services to celestial mechanics as a whole, instead of indicating priority in determining the numerical value of the gravitational constant used in celestial mechanics, as is sometimes considered in referring to his work." Sagitov (1970:713). Gauss (1809) gave the value with nine significant digits, as 3548.18761 arc seconds.
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Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.
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The James–Stein estimator is a biased estimator of the mean of Gaussian random vectors. It can be shown that the James–Stein estimator dominates the "ordinary" least squares approach, i.e., it has lower mean squared error. It is the best-known example of Stein's phenomenon.
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Maximally informative dimensions does not make any assumptions about the Gaussianity of the stimulus set, which is important, because naturalistic stimuli tend to have non-Gaussian statistics. In this way the technique is more robust than other dimensionality reduction techniques such as spike-triggered covariance analyses.
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263 is a prime, safe prime, happy number, sum of five consecutive primes (43 + 47 + 53 + 59 + 61), balanced prime, Chen prime, Eisenstein prime with no imaginary part, strictly non-palindromic number, Bernoulli irregular prime, Euler irregular prime, Gaussian prime, full reptend prime, Solinas prime, Ramanujan prime.
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Siegman showed that all beam profiles — Gaussian, flat top, TEMXY, or any shape — must follow the equation above provided that the beam radius uses the D4σ definition of the beam width. Using the 10/90 knife-edge, the D86, or the FWHM widths does not work.
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We can see this from the beta-function for the coupling parameter, g. Even though the quantized massless φ4 is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the Wilson-Fisher fixed point, below.
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Hilbert's lemma was proposed at the end of the 19th century by mathematician David Hilbert. The lemma describes a property of the principal curvatures of surfaces. It may be used to prove Liebmann's theorem that a compact surface with constant Gaussian curvature must be a sphere..
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SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000. Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco), most famously in the ADF suite of DFT codes. After the work of John Pople, Warren.
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Doyle's early work was in the mathematics of robust control, linear-quadratic-Gaussian control robustness, (structured) singular value analysis, H-infinity. He has coauthored books and software toolboxes, a control analysis tool for high performance commercial and military aerospace systems, as well as other industrial systems.
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This corresponds to the coherence length because the difference of the optical path length is twice the length difference of the reference and measurement arms of the interferometer. The relationship between correlogram width, coherence length and spectral width is calculated for the case of a Gaussian spectrum.
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Shift-register for the (7, [171, 133]) convolutional code polynomial. Branches: h^1 = 171_o = [1111001]_b, h^2 = 133_o = [1011011]_b. All of the math operations should be done by modulo 2. Theoretical bit-error rate curves of encoded QPSK (soft decision), additive white Gaussian noise channel.
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Mainardi became a professor of mathematical physics at Bologna. From 1971 to 1973, he was a lecturer at the Marche Polytechnic University on rational mechanics. He teaches non-Gaussian stochastics and mathematics among other science-related subjects. He has offered courses on statistical mechanics and fractional calculus.
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By construction, the marginal distribution of \tau is a gamma distribution, and the conditional distribution of x given \tau is a Gaussian distribution. The marginal distribution of x is a three-parameter non-standardized Student's t-distribution with parameters ( u, \mu, \sigma^2)=(2\alpha, \mu, \beta/(\lambda\alpha)).
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757, 888-907. M. Wilczek & C. Meneveau, “Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields“ (2014), J. Fluid Mech. 756, 191-225. L. Biferale, C. Meneveau & R. Verzicco, “Deformation statistics of sub-Kolmogorov-scale ellipsoidal drops in isotropic turbulence“ (2014), J. Fluid Mech.
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An important class of point processes, with applications to physics, random matrix theory, and combinatorics, is that of determinantal point processes.Hough, J. B., Krishnapur, M., Peres, Y., and Virág, B., Zeros of Gaussian analytic functions and determinantal point processes. University Lecture Series, 51. American Mathematical Society, Providence, RI, 2009.
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Prominent examples of stochastic algorithms are Markov chains and various uses of Gaussian distributions. Stochastic algorithms are often used together with other algorithms in various decision- making processes. Music has also been composed through natural phenomena. These chaotic models create compositions from the harmonic and inharmonic phenomena of nature.
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A related phenomenon is dithering applied to analog signals before analog-to-digital conversion. Stochastic resonance can be used to measure transmittance amplitudes below an instrument's detection limit. If Gaussian noise is added to a subthreshold (i.e., immeasurable) signal, then it can be brought into a detectable region.
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The mosaic crystal model goes back to a theoretical analysis of X-ray diffraction by C. G. Darwin (1922). Currently, most studies follow Darwin in assuming a Gaussian distribution of crystallite orientations centered on some reference orientation. The mosaicity is commonly equated with the standard deviation of this distribution.
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The cylinder is an example of a developable surface. In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression).
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Alice Guionnet also demonstrated significant results in free probabilities by comparing Voiculescu entropies, building with Vaughan Jones and Dimitri Shlyakhtenko a round of subfactors from planar algebras of any index, and establishing isomorphisms between the algebras of von Neumann generated by q-Gaussian variables by constructing free transport.
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The Moffat distribution, named after the physicist Anthony Moffat, is a continuous probability distribution based upon the Lorentzian distribution. Its particular importance in astrophysics is due to its ability to accurately reconstruct point spread functions, whose wings cannot be accurately portrayed by either a Gaussian or Lorentzian function.
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Scale- Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, pages 148-163, 2003.Crowley, J, Riff O. Fast computation of scale normalised Gaussian receptive fields, Proc. Scale-Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, 2003.
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There are many algorithms for computing the nodes and weights of Gaussian quadrature rules. The most popular are the Golub-Welsch algorithm requiring operations, Newton's method for solving p_n(x) = 0 using the three- term recurrence for evaluation requiring operations, and asymptotic formulas for large n requiring operations.
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Carl Friedrich Gauss (1777–1855) Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective Gaussian is pronounced .
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Given the initial positions (e.g., from theoretical knowledge) and velocities (e.g., randomized Gaussian), we can calculate all future (or past) positions and velocities. One frequent source of confusion is the meaning of temperature in MD. Commonly we have experience with macroscopic temperatures, which involve a huge number of particles.
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The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.
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The Darmois–Skitovich theorem is one of the most famous characterization theorems of mathematical statistics. It characterizes the normal distribution (the Gaussian distribution) by the independence of two linear forms from independent random variables. This theorem was proved independently by G. Darmois and V. P. Skitovich in 1953.
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Let \xi_j, j = 1, 2, \ldots, n, n \ge 2 be independent random variables. Let \alpha_j, \beta_j be nonzero constants. If the linear forms L_1 = \alpha_1\xi_1 + \cdots + \alpha_n\xi_n and L_2 = \beta_1\xi_1 + \cdots + \beta_n\xi_n are independent then all random variables \xi_j have normal distributions (Gaussian distributions).
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For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. Two different hyperbolas will be formed on either side of the tangent plane. These hyperbolas share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic directions.
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More precisely, Gauss observed that if a+bi is a (Gaussian) prime and a–1+bi is divisible by 2+2i, then the number of solutions to the congruence 1=xx+yy+xxyy (mod a+bi), including x=∞, y=±i and x=±i, y=∞, is (a–1)2+b2.
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At higher temperatures, or when the chromophore interacts strongly with the matrix, the probability of multiphonon is high and the phonon side band approximates a Gaussian distribution. The distribution of intensity between the zero-phonon line and the phonon sideband is characterized by the Debye-Waller factor α.
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On the other hand, a scenario, where a non-Gaussian (i.e. nontrivial) fixed point is approached in the UV limit, is referred to as asymptotic safety. Asymptotically safe theories may be well defined at all scales despite being nonrenormalizable in perturbative sense (according to the classical scaling dimensions).
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CGS-emu (or "electromagnetic cgs") units are one of several systems of electromagnetic units within the centimetre gram second system of units; others include CGS- esu, Gaussian units, and Lorentz–Heaviside units. In these other systems, the abcoulomb is not used; CGS-esu and Gaussian units use the statcoulomb is instead, while the Lorentz-Heaviside unit of charge has no specific name. In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in a vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system.
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Buckminsterfullerene, C60 In 1970, John Pople developed the Gaussian program greatly easing computational chemistry calculations.W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Gaussian 70 (Quantum Chemistry Program Exchange, Program No. 237, 1970). In 1971, Yves Chauvin offered an explanation of the reaction mechanism of olefin metathesis reactions.Catalyse de transformation des oléfines par les complexes du tungstène. II. Télomérisation des oléfines cycliques en présence d'oléfines acycliques Die Makromolekulare Chemie Volume 141, Issue 1, Date: 9 February 1971, Pages: 161–176 Par Jean-Louis Hérisson, Yves Chauvin In 1975, Karl Barry Sharpless and his group discovered a stereoselective oxidation reactions including Sharpless epoxidation,Katsuki, T.; Sharpless, K. B. J. Am. Chem. Soc.
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It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value 2π. As an application, a torus has Euler characteristic 0, so its total curvature must also be zero. If the torus carries the ordinary Riemannian metric from its embedding in R3, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting in total curvature 0.
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Namely, one can define a boson sampling model, where a linear optical evolution of input single-photon states is concluded by Gaussian measurements (more specifically, by eight-port homodyne detection that projects each output mode onto a squeezed coherent state). Such a model deals with continuous-variable measurement outcome, which, under certain conditions, is a computationally hard task. Finally, a linear optics platform for implementing a boson sampling experiment where input single-photons undergo an active (non-linear) Gaussian transformation is also available. This setting makes use of a set of two-mode squeezed vacuum states as a prior resource, with no need of single-photon sources or in-line nonlinear amplification medium.
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Transformations described by symplectic matrices play an important role in quantum optics and in continuous-variable quantum information theory. For instance, symplectic matrices can be used to describe Gaussian (Bogoliubov) transformations of a quantum state of light. In turn, the Bloch-Messiah decomposition () means that such an arbitrary Gaussian transformation can be represented as a set of two passive linear-optical interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix D). In fact, one can circumvent the need for such in-line active squeezing transformations if two-mode squeezed vacuum states are available as a prior resource only.
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In statistics and econometrics one often assumes that an observed series of data values is the sum of a series of values generated by a deterministic linear process, depending on certain independent (explanatory) variables, and on a series of random noise values. Then regression analysis is used to infer the parameters of the model process from the observed data, e.g. by ordinary least squares, and to test the null hypothesis that each of the parameters is zero against the alternative hypothesis that it is non-zero. Hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and have the same Gaussian probability distributionin other words, that the noise is Gaussian white (not just white).
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The intraparietal sulcus and the prefrontal cortex, also implicated in number, communicate in approximating number and it was found in both species that the parietal neurons of the IPS had short firing latencies, whereas the frontal neurons had longer firing latencies. This supports the notion that number is first processed in the IPS and, if needed, is then transferred to the associated frontal neurons in the prefrontal cortex for further numerations and applications. Humans displayed Gaussian curves in the tuning curves of approximate magnitude. This aligned with monkeys, displaying a similarly structured mechanism in both species with classic Gaussian curves relative to the increasingly deviant numbers with 16 and 32 as well as habituation.
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If the coefficients of the matrix are exactly given numbers, the column echelon form of the matrix may be computed by Bareiss algorithm more efficiently than with Gaussian elimination. It is even more efficient to use modular arithmetic and Chinese remainder theorem, which reduces the problem to several similar ones over finite fields (this avoids the overhead induced by the non-linearity of the computational complexity of integer multiplication). For coefficients in a finite field, Gaussian elimination works well, but for the large matrices that occur in cryptography and Gröbner basis computation, better algorithms are known, which have roughly the same computational complexity, but are faster and behave better with modern computer hardware.
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Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied. The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance.
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Regularity, sometimes called Myerson's regularity, is a property of probability distributions used in auction theory and revenue management. Examples of distributions that satisfy this condition include Gaussian, uniform, and exponential; some power law distributions also satisfy regularity. Distributions that satisfy the regularity condition are often referred to as "regular distributions".
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A good quality random Gaussian function with the zero mean is commonly the default in LOBPCG to generate the initial approximations. To fix the initial approximations, one can select a fixed seed for the random number generator. In contrast to the Lanczos method, LOBPCG rarely exhibits asymptotic superlinear convergence in practice.
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Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable. See also Variance gamma process#Option pricing.
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The statvolt is a unit of voltage and electrical potential used in the CGS-ESU and gaussian systems of units. In terms of its relation to the SI units, one statvolt corresponds to exactly , i.e. to 299.792458 volts. The statvolt is also defined in the CGS system as 1 erg / statcoulomb.
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Approximations to Bessel beams are made in practice either by focusing a Gaussian beam with an axicon lens to generate a Bessel–Gauss beam, by using axisymmetric diffraction gratings, or by placing a narrow annular aperture in the far field. High order Bessel beams can be generated by spiral diffraction gratings.
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The factor 1/√(4π) is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform. Instead of one also writes . Note that need not exist for every real number , when the defining integral fails to converge.
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Left: original image. Right: image processed with bilateral filter A bilateral filter is a non-linear, edge-preserving, and noise-reducing smoothing filter for images. It replaces the intensity of each pixel with a weighted average of intensity values from nearby pixels. This weight can be based on a Gaussian distribution.
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Moscardini was born 30 October 1961, in Reggio Emilia, Italy. He was awarded a Laurea degree in 1986 and a PhD. in 1989 in Astronomy (Cosmological N-body simulations with non-Gaussian initial conditions) from University of Bologna. Moscardini did post-doctoral work at the University of Brighton from 1990–1991.
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Dr. Aaron D. Wyner (March 17, 1939 - September 29, 1997) was an American information theorist noted for his contributions in coding theory, particularly the Gaussian channel. He lived in South Orange, New Jersey.Burkhart, Ford. "Aaron D. Wyner, 58; Helped Speed Data Around the Globe", The New York Times, October 13, 1997.
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179 is an odd number. 179 is a prime number; that is, it is not divisible by any integer (except for 1 and itself). It is an Eisenstein prime, as it is indivisible even by complex Gaussian integers. It is a Chen prime, being two less than another prime, 181.
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See, for example, cascading gauge theory. Noncommutative quantum field theories have a UV cutoff even though they are not effective field theories. Physicists distinguish between trivial and nontrivial fixed points. If a UV fixed point is trivial (generally known as Gaussian fixed point), the theory is said to be asymptotically free.
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3, pp. 272–298, 2003. demonstrated that the problem of minimizing image and mechanical energy can be reformulated in solving the energy image then applying a Gaussian filter at each iteration. We use this strategy in Yadics and we add the median filter as it is massively used in PIV.
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A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations; see Illustration of the central limit theorem for further details. Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the Central Limit Theorem.
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It can be used to turn a Gaussian beam into a non-diffractive Bessel-like beam. Axicons were first proposed in 1954 by John McLeod. Axicons are used in atomic traps and for generating plasma in wakefield accelerators. They are used in eye surgery in cases where a ring-shaped spot is useful.
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Bayesian optimization of a function (black) with Gaussian processes (purple). Three acquisition functions (blue) are shown at the bottom. Since the objective function is unknown, the Bayesian strategy is to treat it as a random function and place a prior over it. The prior captures beliefs about the behavior of the function.
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The concept of quantum illumination was first introduced by Seth Lloyd and collaborators at MIT in 2008. A theoretical proposal for quantum illumination using Gaussian states was proposed by Jeffrey Shapiro and collaborators. The basic setup of quantum illumination is target detection. Here the sender prepares two entangled systems, called signal and idler.
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Cavallo developed an improvement of the Vickrey–Clarke–Groves mechanism in which money is redistributed in order to increase social welfare. He tested his mechanism using simulations. He generated piecewise-constant valuation functions, whose constants were selected at random from the uniform distribution. He also tried Gaussian distributions and got similar results.
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For a (not necessarily invertible) matrix over any field, the exact necessary and sufficient conditions under which it has an LU factorization are known. The conditions are expressed in terms of the ranks of certain submatrices. The Gaussian elimination algorithm for obtaining LU decomposition has also been extended to this most general case.
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Piscataway, N.J.: IEEE, 2010. Print. 3D Gaussian filters are used to extract orientation measurements. They were chosen due to their ability to capture a broad spectrum and easy and efficient computations.Lee, Kyoung Mu. Computer Vision-- ACCV 2012 11th Asian Conference on Computer Vision, Daejeon, Korea, November 5–9, 2012, Revised Selected Papers.
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Studied by Eugenio Beltrami in 1868, as a surface of constant negative Gaussian curvature, the pseudosphere is a local model of hyperbolic geometry. The idea was carried further by Kasner and Newman in their book Mathematics and the Imagination, where they show a toy train dragging a pocket watch to generate the tractrix.
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Euler made the first conjectures about biquadratic reciprocity.Euler, Tractatus, § 456 Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas.
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84 The numbers built up from a cube root of unity are now called the ring of Eisenstein integers. The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
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The classes of affective intent were then modeled as a gaussian mixture model and trained with these samples using the expectation- maximization algorithm. Classification is done with multiple stages, first classifying an utterance into one of two general groups (e.g. soothing/neutral vs. prohibition/attention/approval) and then doing more detailed classification.
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Find circles in a shoe-print The original picture (right) is first turned into a binary image (left) using a threshold and Gaussian filter. Then edges (mid) are found from it using canny edge detection. After this, all the edge points are used by the Circle Hough Transform to find underlying circle structure.
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A plot of the Q-function. In statistics, the Q-function is the tail distribution function of the standard normal distribution.The Q-function, from cnx.orgBasic properties of the Q-function In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations.
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In differential geometry, the Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature.
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The logical building block for this theory was the use of the Gaussian air pollutant dispersion equation for point sources. www.crcpress.com www.air-dispersion.com One of the early point source air pollutant plume dispersion equations was derived by Bosanquet and PearsonC.H. Bosanquet and J.L. Pearson, "The spread of smoke and gases from chimneys", Trans.
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In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series. It is named after the mathematician and statistician Peter Whittle, who introduced it in his PhD thesis in 1951. It is commonly utilized in time series analysis and signal processing for parameter estimation and signal detection.
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Also, Gaussian weighting provided no benefit when used in conjunction with the C-HOG blocks. C-HOG blocks appear similar to shape context descriptors, but differ strongly in that C-HOG blocks contain cells with several orientation channels, while shape contexts only make use of a single edge presence count in their formulation.
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In this way, discretization effects over space and scale can be reduced to a minimum allowing for potentially more accurate image descriptors. In Lindeberg (2015) such pure Gauss-SIFT image descriptors were combined with a set of generalized scale-space interest points comprising the Laplacian of the Gaussian, the determinant of the Hessian, four new unsigned or signed Hessian feature strength measures as well as Harris-Laplace and Shi-and-Tomasi interests points. In an extensive experimental evaluation on a poster dataset comprising multiple views of 12 posters over scaling transformations up to a factor of 6 and viewing direction variations up to a slant angle of 45 degrees, it was shown that substantial increase in performance of image matching (higher efficiency scores and lower 1-precision scores) could be obtained by replacing Laplacian of Gaussian interest points by determinant of the Hessian interest points. Since difference-of-Gaussians interest points constitute a numerical approximation of Laplacian of the Gaussian interest points, this shows that a substantial increase in matching performance is possible by replacing the difference-of-Gaussians interest points in SIFT by determinant of the Hessian interest points.
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Visualization of a buoyant Gaussian air pollutant dispersion plume The technical literature on air pollution dispersion is quite extensive and dates back to the 1930s and earlier. One of the early air pollutant plume dispersion equations was derived by Bosanquet and Pearson.Bosanquet, C. H. and Pearson, J. L., "The spread of smoke and gases from chimneys", Transactions of the Faraday Society, 32:1249, 1936 Their equation did not assume Gaussian distribution nor did it include the effect of ground reflection of the pollutant plume. Sir Graham Sutton derived an air pollutant plume dispersion equation in 1947 which did include the assumption of Gaussian distribution for the vertical and crosswind dispersion of the plume and also included the effect of ground reflection of the plume.Sutton, O. G., "The problem of diffusion in the lower atmosphere", Quarterly Journal of the Royal Meteorological Society, 73:257, 1947 and "The theoretical distribution of airborne pollution from factory chimneys", Quarterly Journal of the Royal Meteorological Society, 73:426, 1947 Under the stimulus provided by the advent of stringent environmental control regulations, there was an immense growth in the use of air pollutant plume dispersion calculations between the late 1960s and today.
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The fact that the Dirichlet distribution is a probability distribution on the simplex of sets of non-negative numbers that sum to one makes it a good candidate to model distributions over distributions or distributions over functions. Additionally, the nonparametric nature of this model makes it an ideal candidate for clustering problems where the distinct number of clusters is unknown beforehand. In addition, the Dirichlet process has also been used for developing a mixture of expert models, in the context of supervised learning algorithms (regression or classification settings). For instance, mixtures of Gaussian process experts, where the number of required experts must be inferred from the data.Sotirios P. Chatzis, “A Latent Variable Gaussian Process Model with Pitman-Yor Process Priors for Multiclass Classification,” Neurocomputing, vol. 120, pp.
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The phenomenon of revivals is most readily observable for the wave functions being well localized wave packets at the beginning of the time evolution for example in the hydrogen atom. For Hydrogen, the fractional revivals show up as multiple angular Gaussian bumps around the circle drawn by the radial maximum of leading circular state component (that with the highest amplitude in the eigenstate expansion) of the original localized state and the full revival as the original Gaussian . The full revivals are exact for the infinite quantum well, harmonic oscillator or the hydrogen atom, while for shorter times are approximate for the hydrogen atom and a lot of quantum systems. 500px The plot of collapses and revivals of quantum oscillations of the JCM atomic inversion.
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The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle. As early as 1693 Christiaan Huygens found that the volume and the surface area of the pseudosphere are finite,, Chapter 17, page 324 despite the infinite extent of the shape along the axis of rotation. For a given edge radius , the area is just as it is for the sphere, while the volume is and therefore half that of a sphere of that radius.
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Knowledge about an input quantity X_i is inferred from repeated measured values ("Type A evaluation of uncertainty"), or scientific judgement or other information concerning the possible values of the quantity ("Type B evaluation of uncertainty"). In Type A evaluations of measurement uncertainty, the assumption is often made that the distribution best describing an input quantity X given repeated measured values of it (obtained independently) is a Gaussian distribution. X then has expectation equal to the average measured value and standard deviation equal to the standard deviation of the average. When the uncertainty is evaluated from a small number of measured values (regarded as instances of a quantity characterized by a Gaussian distribution), the corresponding distribution can be taken as a t-distribution.
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In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free information per time unit that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley.
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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 about 50%, decreasing to about 23% for an N-dimensional Gaussian target distribution. If \sigma^2 is too small, the chain will mix slowly (i.e., the acceptance rate will be high, but successive samples will move around the space slowly, and the chain will converge only slowly to P(x)). On the other hand, if \sigma^2 is too large, the acceptance rate will be very low because the proposals are likely to land in regions of much lower probability density, so a_1 will be very small, and again the chain will converge very slowly.
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This is called the "joint ratio", and can be used to measure the degree of imbalance: a joint ratio of 96:4 is extremely imbalanced, 80:20 is highly imbalanced (Gini index: 76%), 70:30 is moderately imbalanced (Gini index: 28%), and 55:45 is just slightly imbalanced (Gini index 14%). The Pareto principle is an illustration of a "power law" relationship, which also occurs in phenomena such as brush fires and earthquakes. Because it is self- similar over a wide range of magnitudes, it produces outcomes completely different from Normal or Gaussian distribution phenomena. This fact explains the frequent breakdowns of sophisticated financial instruments, which are modeled on the assumption that a Gaussian relationship is appropriate to something like stock price movements.
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As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of ε, while for larger values of ε it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If ε has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window.
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G4 is a compound method in spirit of the other Gaussian theories and attempts to take the accuracy achieved with G3X one small step further. This involves the introduction of an extrapolation scheme for obtaining basis set limit Hartree-Fock energies, the use of geometries and thermochemical corrections calculated at B3LYP/6-31G(2df,p) level, a highest- level single point calculation at CCSD(T) instead of QCISD(T) level, and addition of extra polarization functions in the largest-basis set MP2 calculations. Thus, Gaussian 4 (G4) theory is an approach for the calculation of energies of molecular species containing first-row (Li–F), second-row (Na–Cl), and third row main group elements. G4 theory is an improved modification of the earlier approach G3 theory.
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For black-body radiation, the phase-space functional is Gaussian. The resulting occupation distribution of the number state is characterized by a Bose–Einstein statistics for which Q=\langle n\rangle .Mandel, L., and Wolf, E., Optical Coherence and Quantum Optics (Cambridge 1995) Coherent states have a Poissonian photon-number statistics for which Q=0 .
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The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade,The Prince of Mathematics. The Door to Science by keplersdiscovery.com. fueled Gauss's interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. Among other things, he came up with the notion of Gaussian curvature.
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The elastix software also offers other features that can be employed to speed up the registration procedure and to provide more advanced algorithms to the end-users. Some examples are the introduction of blur and Gaussian pyramid to reduce data complexity, and multi-image and multi-metric framework to deal with more complex applications.
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An alternative approach is known as the Hartree approximation or self-consistent one-loop approximation (Amit 1984). It takes advantage of Gaussian fluctuation corrections to the 0^{th}-order MF contribution, to renormalize the model parameters and extract in a self-consistent way the dominant length scale of the concentration fluctuations in critical concentration regimes.
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He remained at Cambridge until his death. He was only elected to a Cambridge College Fellowship at University College, now Wolfson College, Cambridge, shortly before his death. Boys is best known for the introduction of Gaussian orbitals into ab initio quantum chemistry. Almost all basis sets used in computational chemistry now employ these orbitals.
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Intermediate levels of Re(f) = constant are shown with thin red lines for negative values and with thin blue lines for positive values. The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf(z) approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.
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Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of in , the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity.
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The Gaussian unit system is just one of several electromagnetic unit systems within CGS. Others include "electrostatic units", "electromagnetic units", and Lorentz–Heaviside units. Some other unit systems are called "natural units", a category that includes Hartree atomic units, Planck units, and others. SI units are by far the most common system of units today.
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It also frequently appears in various integrals involving Gaussian functions. Computer algorithms for the accurate calculation of this function are available;Patefield, M. and Tandy, D. (2000) "Fast and accurate Calculation of Owen’s T-Function", Journal of Statistical Software, 5 (5), 1-25\. quadrature having been employed since the 1970s. JC Young and Christoph Minder.
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When a linear second-order ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Q-value of the equation. Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.
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The model above can be enhanced. A longer, "effective" unit length can be defined such that the chain can be regarded as freely-jointed, along with a smaller N, such that the constraint L = N x l is still obeyed. It, too, gives a Gaussian distribution. However, specific cases can also be precisely calculated.
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A Laplacian pyramid is very similar to a Gaussian pyramid but saves the difference image of the blurred versions between each levels. Only the smallest level is not a difference image to enable reconstruction of the high resolution image using the difference images on higher levels. This technique can be used in image compression.
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In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning.
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Descartes, René, Progymnasmata de solidorum elementis, in Oeuvres de Descartes, vol. X, pp. 265–276 A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron. This is a special case of the Gauss–Bonnet theorem which relates the integral of the Gaussian curvature to the Euler characteristic.
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By Poincaré's uniformization theorem, any orientable closed 2-manifold is conformally equivalent to a surface of constant curvature 0, +1 or –1. In other words, by multiplying the metric by a positive scaling factor, the Gaussian curvature can be made to take exactly one of these values (the sign of the Euler characteristic of ).; .
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Graphical models can still be used when the variables of choice are continuous. In these cases, the probability distribution is represented as a multivariate probability distribution over continuous variables. Each family of distribution will then impose certain properties on the graphical model. Multivariate Gaussian distribution is one of the most convenient distributions in this problem.
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237x237px 237x237px Wavelets are often used to denoise two dimensional signals, such as images. The following example provides three steps to remove unwanted white Gaussian noise from the noisy image shown. Matlab was used to import and filter the image. The first step is to choose a wavelet type, and a level N of decomposition.
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Another statistical measurement is defined for evaluating network motifs, but it is rarely used in known algorithms. This measurement is introduced by Picard et al. in 2008 and used the Poisson distribution, rather than the Gaussian normal distribution that is implicitly being used above. In addition, three specific concepts of sub-graph frequency have been proposed.
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In all the formulas stated below the sides , , and must be measured in absolute length, a unit so that the Gaussian curvature of the plane is −1. In other words, the quantity in the paragraph above is supposed to be equal to 1. Trigonometric formulas for hyperbolic triangles depend on the hyperbolic functions sinh, cosh, and tanh.
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Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. See also and .
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This problem can be solved by introducing a cluster-expansion transformation (CET)Kira, M.; Koch, S. (2008). "Cluster-expansion representation in quantum optics". Physical Review A 78 (2). doi:10.1103/PhysRevA.78.022102 that represents the distribution in terms of a Gaussian, defined by the singlet–doublet contributions, multiplied by a polynomial, defined by the higher-order clusters.
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Like other potentials, many different electromagnetic four- potentials correspond to the same electromagnetic field, depending upon the choice of gauge. This article uses tensor index notation and the Minkowski metric sign convention . See also covariance and contravariance of vectors and raising and lowering indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units.
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39 (2011), no. 3, 259–292. Utilizing an argument of Perelman's, Cao and Detang Zhou showed that complete gradient shrinking Ricci solitons have a Gaussian character, in that for any given point of , the function must grow quadratically with the distance function to . Additionally, the volume of geodesic balls around can grow at most polynomially with their radius.
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If the marginal distributions F_i^{-1}(u_i) are continuous, it follows that C is unique. For properties and proofs of equation (11), see Sklar (1959) and Nelsen (2006). Numerous types of copula functions exist. They can be broadly categorized in one-parameter copulas as the Gaussian copula, and the Archimedean copula, which comprise Gumbel, Clayton and Frank copulas.
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A DUDE-based framework for grayscale image denoising achieves state-of-the-art denoising for impulse-type noise channels (e.g., "salt and pepper" or "M-ary symmetric" noise), and good performance on the Gaussian channel (comparable to the Non-local means image denoising scheme on this channel). A different DUDE variant applicable to grayscale images is presented in.
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The property is set on a container element or on a graphics element to apply a filter effect to it. Each element contains a set of filter primitives as its children. Each filter primitive performs a single fundamental graphical operation (e.g., a Gaussian blur or a lighting effect) on one or more inputs, producing a graphical result.
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For example, a generalization of Gaussian elimination called Buchberger's algorithm has for its complexity an exponential function of the problem data (the degree of the polynomials and the number of variables of the multivariate polynomials). Because exponential functions eventually grow much faster than polynomial functions, an exponential complexity implies that an algorithm has slow performance on large problems.
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Stein's example is an important result in decision theory which can be stated as : The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk in dimension at least 3. The following is an outline of its proof. The reader is referred to the main article for more information.
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The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To distinguish the two families, they are referred to below as "version 1" and "version 2". However this is not a standard nomenclature.
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The Wyner–Ziv theorem presents the achievable lower bound for the bit rate of X at given distortion D. It was found that for Gaussian memoryless sources and mean- squared error distortion, the lower bound for the bit rate of X remain the same no matter whether side information is available at the encoder or not.
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Lag windowing is a technique that consists of windowing the autocorrelation coefficients prior to estimating linear prediction coefficients (LPC). The windowing in the autocorrelation domain has the same effect as a convolution (smoothing) in the power spectral domain and helps in stabilizing the result of the Levinson-Durbin algorithm. The window function is typically a Gaussian function.
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In the early 1960s, the perturbation theory in quantum chemical applications was introduced. Since, there has been a wide spread of uses of the theory through software such as Gaussian. The perturbation theory correlation method is used routinely by the non-specialists. This is because it can easily achieve the property of size extensivity comparing to other correlation methods.
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Like Gaussian processes, and unlike SVMs, RBF networks are typically trained in a maximum likelihood framework by maximizing the probability (minimizing the error). SVMs avoid overfitting by maximizing instead a margin. SVMs outperform RBF networks in most classification applications. In regression applications they can be competitive when the dimensionality of the input space is relatively small.
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The definition of "quality" also depends on the application. While a high-quality single-mode Gaussian beam (M2 close to unity) is optimum for many applications, for other applications a uniform multimode tophat beam intensity distribution is required. An example is laser surgery. Power-in-the-bucket and Strehl ratio are two other attempts to define beam quality.
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In that paper, the Gaussian chirplet transform was presented as one such example, together with a successful application to ice fragment detection in radar (improving target detection results over previous approaches). The term chirplet (but not the term chirplet transform) was also proposed for a similar transform, apparently independently, by Mihovilovic and Bracewell later that same year.
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LPC coefficients, MFCC) then models them using a Gaussian mixture model (GMM). After a model is obtained using the data collected, conditional probability is formed for each target contained in the training database. In this example, there are M blocks of data. This will result in a collection of M probabilities for each target in the database.
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This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae.
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This method can also slightly correct the effect of the non-Gaussian noise that has caused trouble to accurately identify the states using the statistical methods. The current data analysis for smFRET still requires great care and special training which is in a call for deep-learning algorithms to play a role to free the labor in data analysis.
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Solution of a 1D heat partial differential equation. The temperature (u) is initially distributed over a one-dimensional, one-unit-long interval (x = [0,1]) with insulated endpoints. The distribution approaches equilibrium over time. The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution).
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Therefore, :(p_{r+1},p_s)=(xp_r,p_s)-a_{r,s}(p_s,p_s)=(xp_r,p_s)-(xp_r,p_s)=0. However, if the scalar product satisfies (xf, g) = (f,xg) (which is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation: For s < r - 1, xp_s is a polynomial of degree less than or equal to .
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Free, massless quantized scalar field theory has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the Gaussian fixed point. However, even though the classical massless φ4 theory is scale-invariant in D=4, the quantized version is not scale-invariant.
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This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.
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Consequently, the Gaussian theory only supplies a convenient method of approximating reality; realistic optical systems fall short of this unattainable ideal. Currently, all that can be accomplished it the projection of a single plane onto another plane; but even in this, aberrations always occurs and it may be unlikely that these will ever be entirely corrected.
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In his posthumously published Kollektivmasslehre (1897), Fechner introduced the Zweiseitige Gauss'sche Gesetz or two-piece normal distribution, to accommodate the asymmetries he had observed in empirical frequency distributions in many fields. The distribution has been independently rediscovered by several authors working in different fields.Wallis, K.F. (2014). "The two-piece normal, binormal, or double Gaussian distribution: its origin and rediscoveries".
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Abraham Wald re-derived this distribution in 1944 as the limiting form of a sample in a sequential probability ratio test. The name inverse Gaussian was proposed by Maurice Tweedie in 1945. Tweedie investigated this distribution in 1956 and 1957 and established some of its statistical properties. The distribution was extensively reviewed by Folks and Chhikara in 1978.
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It follows that is a complete metric of constant curvature 0 on the complement of , which is therefore isometric to the plane. Composing with stereographic projection, it follows that there is a smooth function such that has Gaussian curvature +1 on the complement of . The function automatically extends to a smooth function on the whole of .
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The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger than the arc-length of a horocycle connecting two points. If the Gaussian curvature of the plane is −1 then the geodesic curvature of a horocycle is 1 and of a hypercycle is between 0 and 1.
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Some, like height for a given sex, vary in close to a "normal" or Gaussian distribution. Other characteristics (e.g., skin color) vary continuously in a population, but the continuum may be socially divided into a small number of distinct categories. Then, there are some characteristics that vary bimodally (for example, handedness), with fewer people in intermediate categories.
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This section expands on the correspondence between infinitely wide neural networks and Gaussian processes for the specific case of a fully connected architecture. It provides a proof sketch outlining why the correspondence holds, and introduces the specific functional form of the NNGP for fully connected networks. The proof sketch closely follows the approach in Novak, et al., 2018.
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The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was invented by Charles Dupin. For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse aligned with the principal directions.
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In this regime, the density contrast field is Gaussian, Fourier modes evolve independently, and the power spectrum is sufficient to completely describe the density field. On small scales, gravitational collapse is non- linear, and can only be computed accurately using N-body simulations. Higher- order statistics are necessary to describe the full field at small scales.
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In order to properly describe electronic delocalized states, a previously optimized standard basis set can be complemented with additional delocalized Gaussian functions with small exponent values, generated by the even-tempered scheme. This approach has also been employed to generate basis sets for other types of quantum particles rather than electrons, like quantum nuclei, negative muons or positrons.
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In mathematics, symmetric convolution is a special subset of convolution operations in which the convolution kernel is symmetric across its zero point. Many common convolution-based processes such as Gaussian blur and taking the derivative of a signal in frequency-space are symmetric and this property can be exploited to make these convolutions easier to evaluate.
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Gauss himself stated the constant in arc seconds, with nine significant digits, as . In the late 19th century, this value was adopted, and converted to radian, by Simon Newcomb, as . and the constant appears in this form in his Tables of the Sun, published in 1898. "The adopted value of the Gaussian constant is that of Gauss himself, namely: ".
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Random Jitter, also called Gaussian jitter, is unpredictable electronic timing noise. Random jitter typically follows a normal distribution due to being caused by thermal noise in an electrical circuit or due to the central limit theorem. The central limit theorem states that composite effect of many uncorrelated noise sources, regardless of the distributions, approaches a normal distribution.
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David X. Li ( born Nanjing, China in the 1960s) is a Chinese-born Canadian quantitative analyst and actuary who pioneered the use of Gaussian copula models for the pricing of collateralized debt obligations (CDOs) in the early 2000s. The Financial Times has called him "the world’s most influential actuary", while in the aftermath of the global financial crisis of 2008–2009, to which Li's model has been partly credited to blame, his model has been called a "recipe for disaster" in the hands of those who did not fully understand his research. Widespread application of simplified Gaussian copula models to financial products such as securities may have contributed to the global financial crisis of 2008–2009. David Li is currently an Adjunct Professor at the University of Waterloo in the Statistics and Actuarial Sciences department.
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Back in the Alcubierre model, it is notable that an outside observer would perceive a uniform potential (from the uniform boost within the sphere) that represents the warp-field region even though it originates from a toroidal energy density. It has similar characteristics as a Gaussian spherical surface held at constant electrostatic potential. To the outside observer, the warp-field sphere has a uniform energy density. By expanding the spherical region while maintaining the same relative boost value for the Gaussian surface, when considering the first law of thermodynamics, the following can be concluded (limited to the 3+1 brane): dE can be replaced by \rho_sdV, which is the total energy for the warp sphere with the same volume change dV as on the right side of the equation.
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The residue class ring modulo a Gaussian integer is a field if and only if z_0 is a Gaussian prime. If is a decomposed prime or the ramified prime (that is, if its norm is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, ). It is thus isomorphic to the field of the integers modulo . If, on the other hand, is an inert prime (that is, is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has elements, and it is an extension of degree 2 (unique, up to an isomorphism) of the prime field with elements (the integers modulo ).
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If the resolution is not limited by the rectangular sampling rate of either the source or target image, then one should ideally use rotationally symmetrical filter or interpolation functions, as though the data were a two dimensional function of continuous x and y. The sinc function of the radius has too long a tail to make a good filter (it is not even square-integrable). A more appropriate analog to the one-dimensional sinc is the two-dimensional Airy disc amplitude, the 2D Fourier transform of a circular region in 2D frequency space, as opposed to a square region. Gaussian plus differential function One might consider a Gaussian plus enough of its second derivative to flatten the top (in the frequency domain) or sharpen it up (in the spatial domain), as shown.
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The infinite-dimensional case raises subtle mathematical issues; we will consider here the finite-dimensional case. We start with a brief review of the main ideas underlying kernel methods for scalar learning, and briefly introduce the concepts of regularization and Gaussian processes. We then show how both points of view arrive at essentially equivalent estimators, and show the connection that ties them together.
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Realization of white Gaussian noise (a) and harmonic oscillation (c), together with the dependencies of the corresponding sample mean on the averaging interval (b, d) Example 2. Fig. 2a and Fig. 2b show how the mains voltage in a city fluctuates quickly, while the average changes slowly. As the averaging interval increases from zero to one hour, the average voltage stabilizes (Fig.
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The most common used model based approach in signal processing is the maximum likelihood (ML) technique. This method requires a statistical framework for the data generation process. When applying the ML technique to the array processing problem, two main methods have been considered depending on the signal data model assumption. According to the Stochastic ML, the signals are modeled as Gaussian random processes.
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It is also part of the AMPAC, GAMESS (US), PC GAMESS, GAMESS (UK), Gaussian, ORCA and CP2K programs. Later, it was essentially replaced by two new methods, PM3 and AM1, which are similar but have different parameterisation methods. The extension by W. Thiel's group, called MNDO/d, which adds d functions, is widely used for organometallic compounds. It is included in GAMESS (UK).
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This led to the study of unique factorization domains, which generalize what was just illustrated in the integers. Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in but it is not in , the ring of Gaussian integers, since and 2 does not divide any factor on the right.
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In the ring of integers (on real numbers), if is a unit, then is either 2 or 0. But are the usual Mersenne primes, and the formula does not lead to anything interesting (since it is always −1 for all ). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.
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An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm.
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Cramer's rule is useful for reasoning about the solution, but, except for or , it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm. The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense, since this determinant is independent of the choice of the basis.
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Leibniz arranged the coefficients of a system of linear equations into an array, now called a matrix, in order to find a solution to the system if it existed. This method was later called Gaussian elimination. Leibniz laid down the foundations and theory of determinants, although Seki Takakazu discovered determinants well before Leibniz. His works show calculating the determinants using cofactors.
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It is the fundamental transverse mode of the laser resonator and has the same form as a Gaussian beam. The pattern has a single lobe, and has a constant phase across the mode. Modes with increasing show concentric rings of intensity, and modes with increasing show angularly distributed lobes. In general there are spots in the mode pattern (except for ).
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The gauss, symbol (sometimes Gs), is a unit of measurement of magnetic induction, also known as magnetic flux density. The unit is part of the Gaussian system of units, which inherited it from the older CGS-EMU system. It was named after the German mathematician and physicist Carl Friedrich Gauss in 1936. One gauss is defined as one maxwell per square centimeter.
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The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.
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Stimulus-response associations may be both encoded and decoded in one non-iterative transformation. The mathematical basis requires no optimization of parameters or error backpropagation, unlike connectionist neural networks. The principal requirement is for stimulus patterns to be made symmetric or orthogonal in the complex domain. HAM typically employs sigmoid pre-processing where raw inputs are orthogonalized and converted to Gaussian distributions.
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Blender has a node-based compositor within the rendering pipeline accelerated with OpenCL.The Video Sequence Editor (VSE) Blender also includes a non-linear video editor called the Video Sequence Editor (VSE), with support for effects like Gaussian blur, color grading, fade and wipe transitions, and other video transformations. However, there is no multi-core support for rendering video with the VSE.
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In the past several years, based on the statistical algorithm development by Lawrence and his collaborators, several programs have also been publicly available and widely used, such as the Gibbs Motif Sampler, the Bayes aligner, Sfold, BALSA, Gibbs Gaussian Clustering, and Bayesian Motif Clustering. His work in Bayesian Statistics won the Mitchell Prize for outstanding applied Bayesian statistics paper in 2000.
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The HLU quantity LH describing a charge is then larger than the corresponding Gaussian quantity (see below), and the rest follows. When dimensional analysis for SI units is used, including and are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is . When one puts , , , and second, this evaluates as .
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Time is another selection criteria because many algorithms are iterative and therefore rather slow. The most straightforward way to remove the halftone patterns is the application of a low-pass filter either in spatial or frequency domain. A simple example is a Gaussian filter. It discards the high-frequency information which blurs the image and simultaneously reduces the halftone pattern.
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This led to the question being posed: is it possible to construct all regular polygons with compass and straightedge? If not, which n-gons (that is polygons with n edges) are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae.
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Li graduated from Tsinghua University in 1986, with a bachelor's degree in computer science. She moved to the United States for graduate study, earning a master's degree from Pennsylvania State University in 1990 and a Ph.D. in computer science from the University of California, Berkeley in 1996. Her doctoral dissertation, Sparse Gaussian Elimination on High Performance Computers, was supervised by James Demmel.
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Example priors. In a 'full' model, left, a parameter has a Gaussian prior with mean 0 and standard deviation 0.5. In a 'reduced' model, right, the same parameter has prior mean zero and standard deviation 1/1000. Bayesian model reduction enables the evidence and parameter(s) of the reduced model to be derived from the evidence and parameter(s) of the full model.
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The \zeta(q) are statistically interpreted, as they characterize the evolution of the distributions of the T_X(a) as a goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models. Modelling as a multiplicative cascade also leads to estimation of multifractal properties. This methods works reasonably well, even for relatively small datasets.
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The astronomical unit of time is a time interval of one day (D) of 86400 seconds. The astronomical unit of mass is the mass of the Sun (S). The astronomical unit of length is that length (A) for which the Gaussian gravitational constant (k) takes the value when the units of measurement are the astronomical units of length, mass and time.
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In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207.
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Radial basis functions are functions that have a distance criterion with respect to a center. Radial basis functions have been applied as a replacement for the sigmoidal hidden layer transfer characteristic in multi-layer perceptrons. RBF networks have two layers: In the first, input is mapped onto each RBF in the 'hidden' layer. The RBF chosen is usually a Gaussian.
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Turk and Pentland combined the conceptual approach of the Karhunen–Loève theorem and factor analysis, to develop a linear model. Eigenfaces are determined based on global and orthogonal features in human faces. These features are established in an unsupervised machine learning process with the help of the Gaussian blur. A human face is calculated as a weighted combination of a number of Eigenfaces.
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Some models use fixed bonds, defined at the start of the simulation, while others have dynamic bonding. More recent efforts strive for robust, transferable models with generic functional forms: spherical harmonics, Gaussian kernels, and neural networks. In addition, MD can be used to simulate groupings of atoms within generic particles, called coarse-grained modeling, e.g. creating one particle per monomer within a polymer.
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A similar effect is available for peak functions. 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.
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Demonstration of density estimation using Kernel density estimation: The true density is mixture of two Gaussians centered around 0 and 3, shown with solid blue curve. In each frame, 100 samples are generated from the distribution, shown in red. Centered on each sample, a Gaussian kernel is drawn in gray. Averaging the Gaussians yields the density estimate shown in the dashed black curve.
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Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few , and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase- space regions smaller than , and thus renders such "negative probabilities" less paradoxical.
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QEG flow diagram for the Einstein–Hilbert truncation. Arrows point from UV to IR scales. Dark background color indicates a region of fast flow, in regions of light background the flow is slow or even zero. The latter case includes a vicinity of the Gaussian fixed point in the origin, and the NGFP in the center of the spiralling arrows, respectively.
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The theorem appeared in the second edition of The Doctrine of Chances by Abraham de Moivre, published in 1738. Although de Moivre did not use the term "Bernoulli trials", he wrote about the probability distribution of the number of times "heads" appears when a coin is tossed 3600 times. This is one derivation of the particular Gaussian function used in the normal distribution.
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These zeros play an important role in numerical integration based on Gaussian quadrature. The specific quadrature based on the P_n's is known as Gauss-Legendre quadrature. From this property and the facts that P_n(\pm 1) e 0 , it follows that P_n(x) has n-1 local minima and maxima in (-1,1) . Equivalently, dP_n(x)/dx has n -1 zeros in (-1,1) .
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In the theory of random matrices, the circular ensembles are measures on spaces of unitary matrices introduced by Freeman Dyson as modifications of the Gaussian matrix ensembles. The three main examples are the circular orthogonal ensemble (COE) on symmetric unitary matrices, the circular unitary ensemble (CUE) on unitary matrices, and the circular symplectic ensemble (CSE) on self dual unitary quaternionic matrices.
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Caustic engineering describes the process of solving the inverse problem to computer graphics. That is, given a specific image, to determine a surface whose refracted or reflected light forms this image. In the discrete version of this problem, the surface is divided into several micro-surfaces which are assumed smooth, i.e. the light reflected/refracted by each micro-surface forms a Gaussian caustic.
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Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient methodHestenes, Magnus R.; Stiefel, Eduard (December 1952). "Methods of Conjugate Gradients for Solving Linear Systems".
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This is a comparison of statistical analysis software that allows doing inference with Gaussian processes often using approximations. This article is written from the point of view of Bayesian statistics, which may use a terminology different from the one commonly used in kriging. The next section should clarify the mathematical/computational meaning of the information provided in the table independently of contextual terminology.
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Geometric optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behaviour then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.
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Furthermore, asymptotic safety provides the possibility of inflation without the need of an inflaton field (while driven by the cosmological constant). It was reasoned that the scale invariance related to the non-Gaussian fixed point underlying asymptotic safety is responsible for the near scale invariance of the primordial density perturbations. Using different methods, asymptotically safe inflation was analyzed further by Weinberg.
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Weapons in the game range from shotguns and assault rifles to high-tech gaussian guns and plasma rifles. Gadgets available include cloaking devices, hover boards and implosion grenades. Some weapons or gadgets are upgradeable. The medkit can be upgraded several times, allowing the player to carry substantially more health and increasing their durability, allowing them to stand up to much more powerful enemies.
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In microscope image processing and astronomy, knowing the PSF of the measuring device is very important for restoring the (original) object with deconvolution. For the case of laser beams, the PSF can be mathematically modeled using the concepts of Gaussian beams. For instance, deconvolution of the mathematically modeled PSF and the image, improves visibility of features and removes imaging noise.
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Graduated optimization is a global optimization technique that attempts to solve a difficult optimization problem by initially solving a greatly simplified problem, and progressively transforming that problem (while optimizing) until it is equivalent to the difficult optimization problem.Hossein Mobahi, John W. Fisher III. On the Link Between Gaussian Homotopy Continuation and Convex Envelopes, In Lecture Notes in Computer Science (EMMCVPR 2015), Springer, 2015.
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Speech denoising has been a long lasting problem in audio signal processing. There are many algorithms for denoising if the noise is stationary. For example, the Wiener filter is suitable for additive Gaussian noise. However, if the noise is non- stationary, the classical denoising algorithms usually have poor performance because the statistical information of the non-stationary noise is difficult to estimate.
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Only the Gaussian function is both separable and isotropic. The separable forms of all other window functions have corners that depend on the choice of the coordinate axes. The isotropy/anisotropy of a two-dimensional window function is shared by its two- dimensional Fourier transform. The difference between the separable and radial forms is akin to the result of diffraction from rectangular vs.
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Both sinusoids suffer less SNR loss under the Hann window than under the Blackman–Harris window. In general (as mentioned earlier), this is a deterrent to using high-dynamic-range windows in low-dynamic-range applications. Figure 4: Two different ways to generate an 8-point Gaussian window sequence (σ=0.4) for spectral analysis applications. MATLAB calls them "symmetric" and "periodic".
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Cohn-Vossen's inequality states that in every complete Riemannian 2-manifold S with finite total curvature and finite Euler characteristic, we haveRobert Osserman, A Survey of Minimal Surfaces, Courier Dover Publications, 2002, page 86. : \iint_S K \, dA \le 2\pi\chi(S), where K is the Gaussian curvature, dA is the element of area, and χ is the Euler characteristic.
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This formalises the classical theory of the "moving frame", favoured by French authors. Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields.
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The used modulation is GMSK modulation (Gaussian Minimum Shift Keying). GSM-R is a TDMA (“Time Division Multiple Access”) system. Data transmission is made of periodical TDMA frames (with a period of 4.615 ms), for each carrier frequency (physical channel). Each TDMA frame is divided in 8 time slots, named logical channels (577 µs long, each time-slot), carrying 148 bits of information.
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It created mathematical proof for the Pythagorean theorem, and a mathematical formula for Gaussian elimination. The treatise also provides values of π, which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724, as well as 3.162 by taking the square root of 10.
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The Advanced Systems Analysis Program (ASAP) is optical engineering software used to simulate optical systems. ASAP can handle coherent as well as incoherent light sources. It is a non-sequential ray tracing tool which means that it can be used not only to analyze lens systems but also for stray light analysis. It uses a Gaussian beam approximation for analysis of coherent sources.
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A cylinder set measure on the dual of a nuclear Fréchet space automatically extends to a measure if its Fourier transform is continuous. Example: Let S be the space of Schwartz functions on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space H of L2 functions, which is in turn contained in the space of tempered distributions S′, the dual of the nuclear Fréchet space S: :S \subseteq H \subseteq S'. The Gaussian cylinder set measure on H gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions, S′. The Hilbert space H has measure 0 in S′, by the first argument used above to show that the canonical Gaussian cylinder set measure on H does not extend to a measure on H.
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In the hidden Markov models considered above, the state space of the hidden variables is discrete, while the observations themselves can either be discrete (typically generated from a categorical distribution) or continuous (typically from a Gaussian distribution). Hidden Markov models can also be generalized to allow continuous state spaces. Examples of such models are those where the Markov process over hidden variables is a linear dynamical system, with a linear relationship among related variables and where all hidden and observed variables follow a Gaussian distribution. In simple cases, such as the linear dynamical system just mentioned, exact inference is tractable (in this case, using the Kalman filter); however, in general, exact inference in HMMs with continuous latent variables is infeasible, and approximate methods must be used, such as the extended Kalman filter or the particle filter.
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The ensemble Kalman filter (EnKF) is a Monte Carlo implementation of the Bayesian update problem: given a probability density function (pdf) of the state of the modeled system (the prior, called often the forecast in geosciences) and the data likelihood, Bayes' theorem is used to obtain the pdf after the data likelihood has been taken into account (the posterior, often called the analysis). This is called a Bayesian update. The Bayesian update is combined with advancing the model in time, incorporating new data from time to time. The original Kalman filter, introduced in 1960, assumes that all pdfs are Gaussian (the Gaussian assumption) and provides algebraic formulas for the change of the mean and the covariance matrix by the Bayesian update, as well as a formula for advancing the covariance matrix in time provided the system is linear.
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This approach, developed at the University of North Texas by Angela K. Wilson's research group, utilizes the correlation consistent basis sets developed by Dunning and co-workers. Unlike the Gaussian-n methods, ccCA does not contain any empirically fitted term. The B3LYP density functional method with the cc-pVTZ basis set, and cc-pV(T+d)Z for third row elements (Na - Ar), are used to determine the equilibrium geometry. Single point calculations are then used to find the reference energy and additional contributions to the energy. The total ccCA energy for main group is calculated by: :EccCA = EMP2/CBS \+ ΔECC \+ ΔECV \+ ΔESR \+ ΔEZPE \+ ΔESO The reference energy EMP2/CBS is the MP2/aug-cc-pVnZ (where n=D,T,Q) energies extrapolated at the complete basis set limit by the Peterson mixed gaussian exponential extrapolation scheme.
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Rachev earned a MSc degree from the Faculty of Mathematics at Sofia University in 1974, a PhD degree from Lomonosov Moscow State University under the supervision of Vladimir Zolotarev in 1979, and a Dr Sci degree from Steklov Mathematical Institute in 1986 under the supervision of Leonid Kantorovich, a Nobel Prize winner in economic sciences, Andrey Kolmogorov and Yuri Prokhorov. Currently, he is Professor Emeritus at University of California Santa Barbara, professor of mathematics at Texas Tech University. In mathematical finance, Rachev is known for his work on application of non-Gaussian models for risk assessment, option pricing, and the applications of such models in portfolio theory. He is also known for the introduction of a new risk-return ratio, the "Rachev Ratio", designed to measure the reward potential relative to tail risk in a non-Gaussian setting.
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In subsequent works Baeurle et al. (Baeurle 2002, Baeurle 2002a) applied the concept of tadpole renormalization, which originates from quantum field theory and leads to the Gaussian equivalent representation of the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble. They could convincingly demonstrate that this strategy provides an additional boost in the statistical convergence of the desired ensemble averages (Baeurle 2002).
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It uses mostly gaussian mutation and blending/averaging crossover. Genetic programming (GP) pioneered tree-like representations and developed genetic operators suitable for such representations. Tree-like representations are used in GP to represent and evolve functional programs with desired properties.A Representation for the Adaptive Generation of Simple Sequential Programs , Nichael Lynn Cramer, Proceedings of an International Conference on Genetic Algorithms and their Applications (1985), pp.
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At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened.
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"Quantum spectroscopy with Schrödinger-cat states". Nature Physics 7 (10): 799–804. doi:10.1038/nphys2091 This property is largely based on CET's ability to describe any distribution in the form where a Gaussian is multiplied by a polynomial factor. This technique is already being used to access and derive quantum-optical spectroscopy from a set of classical spectroscopy measurements, which can be performed using high-quality lasers.
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For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix with all diagonal entries equal to one, an upper triangular matrix , and a permutation matrix ; this is a matrix formulation of Gaussian elimination.
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These methods never outperformed non-uniform internal-handcrafting Gaussian mixture model/Hidden Markov model (GMM-HMM) technology based on generative models of speech trained discriminatively. Key difficulties have been analyzed, including gradient diminishing and weak temporal correlation structure in neural predictive models. Additional difficulties were the lack of training data and limited computing power. Most speech recognition researchers moved away from neural nets to pursue generative modeling.
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He joined with G. Boyd, to introduce the concept of Hermite-Gaussian modes into resonator study, influencing all subsequent research conducted on laser resonators. In his work with R.L. Fork and O.E. Martinez in 1994, a mechanism for generating tunable negative dispersion using pairs of prisms was proposed. This invention was instrumental in achieving ultra-short laser pulses, critical in many applications using laser technology.
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In subsequent works Baeurle et al. (Baeurle 2002, Baeurle 2002a, Baeurle 2003, Baeurle 2003a, Baeurle 2004) applied the concept of tadpole renormalization, leading to the Gaussian equivalent representationof the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble. They could convincingly demonstrate that this strategy provides a further boost in the statistical convergence of the desired ensemble averages (Baeurle 2002).
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In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two- dimensional hyperbolic plane. In the descriptions below the constant Gaussian curvature of the plane is −1.
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Assume that q(\mu,\tau) = q(\mu)q(\tau), i.e. that the posterior distribution factorizes into independent factors for \mu and \tau. This type of assumption underlies the variational Bayesian method. The true posterior distribution does not in fact factor this way (in fact, in this simple case, it is known to be a Gaussian-gamma distribution), and hence the result we obtain will be an approximation.
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Preconditioners are useful in iterative methods to solve a linear system Ax=b for x since the rate of convergence for most iterative linear solvers increases because the condition number of a matrix decreases as a result of preconditioning. Preconditioned iterative solvers typically outperform direct solvers, e.g., Gaussian elimination, for large, especially for sparse, matrices. Iterative solvers can be used as matrix-free methods, i.e.
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Broadcast is a collective communication primitive in parallel programming to distribute programming instructions or data to nodes in a cluster it is the reverse operation of reduce. The broadcast operation is widely used in parallel algorithms, such as matrix-vector multiplication, Gaussian elimination and shortest paths. The Message Passing Interface implements broadcast in `MPI_Bcast`.MPI: A Message-Passing Interface StandardVersion 3.0, Message Passing Interface Forum, pp.
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Another method for removing noise is to evolve the image under a smoothing partial differential equation similar to the heat equation, which is called anisotropic diffusion. With a spatially constant diffusion coefficient, this is equivalent to the heat equation or linear Gaussian filtering, but with a diffusion coefficient designed to detect edges, the noise can be removed without blurring the edges of the image.
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In statistics, the graphical lasso is a sparse penalized maximum likelihood estimator for the concentration or precision matrix (inverse of covariance matrix) of a multivariate elliptical distribution. The original variant was formulated to solve Dempster's covariance selection problem for the multivariate Gaussian distribution when observations were limited. Subsequently, the optimization algorithms to solve this problem were improved and extended to other types of estimators and distributions.
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In general, any conjugate prior can be collapsed out, if its only children have distributions conjugate to it. The relevant math is discussed in the article on compound distributions. If there is only one child node, the result will often assume a known distribution. For example, collapsing an inverse-gamma-distributed variance out of a network with a single Gaussian child will yield a Student's t-distribution.
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This section has a list of the basic formulae of electromagnetism, given in both Gaussian and SI units. Most symbol names are not given; for complete explanations and definitions, please click to the appropriate dedicated article for each equation. A simple conversion scheme for use when tables are not available may be found in Ref.A. Garg, "Classical Electrodynamics in a Nutshell" (Princeton University Press, 2012).
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In her research, she uses the star oscillations to determine the internal rotation profile of stars. The oscillations are obtained from both ground and space-based telescopes. In her PROSPERITY project, she used data obtained from the CoRoT satellite and the NASA Kepler satellite. She is currently the Belgian principal investigator on the PLATO mission, Aerts developed methodology using Gaussian mixture classification to analyse the data.
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The voltage pulses produced for every gamma ray that interacts within the detector volume are then analyzed by a multichannel analyzer (MCA). It takes the transient voltage signal and reshapes it into a Gaussian or trapezoidal shape. From this shape, the signal is then converted into a digital form. In some systems, the analog-to-digital conversion is performed before the peak is reshaped.
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Scheimpflug (1904) referenced this concept in his British patent; Carpentier (1901) also described the concept in an earlier British patent for a perspective-correcting photographic enlarger. The concept can be inferred from a theorem in projective geometry of Gérard Desargues; the principle also readily derives from simple geometric considerations and application of the Gaussian thin-lens formula, as shown in the section Proof of the Scheimpflug principle.
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This risk- expected return relationship of efficient portfolios is graphically represented by a curve known as the efficient frontier. All efficient portfolios, each represented by a point on the efficient frontier, are well- diversified. While ignoring higher moments can lead to significant over- investment in risky securities, especially when volatility is high, the optimization of portfolios when return distributions are non-Gaussian is mathematically challenging.
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Similar to temporal phasor the Fourier transformation of spectra can be used to make phasor. Considering a Gaussian spectrum with zero spectral width and changing the emission maximum from channel zero to K the phasor rotates on a circle from small angles to larger angles. This corresponds to shift theorem of Fourier transformation. Changing the spectral width from zero to infinity moves the phasor toward center.
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Several trials were then conducted with noise of increasing amplitude variance. Extracellular recordings were made of the mechanoreceptor response from the extracted nerve. The encoding of the pressure stimulus in the neural signal was measured by the coherence of the stimulus and response. The coherence was found to be maximized by a particular level of input Gaussian noise, consistent with the occurrence of stochastic resonance.
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Thus expressing complexities in terms of \omega provide a more realistic complexity, since it remains valid whichever algorithm is chosen for matrix computation. Problems that have the same asymptotic complexity as matrix multiplication include determinant, matrix inversion, Gaussian elimination (see next section). Problems with complexity that is expressible in terms of \omega include characteristic polynomial, eigenvalues (but not eigenvectors), Hermite normal form, and Smith normal form.
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That the ICA separation of mixed signals gives very good results is based on two assumptions and three effects of mixing source signals. Two assumptions: #The source signals are independent of each other. #The values in each source signal have non-Gaussian distributions. Three effects of mixing source signals: #Independence: As per assumption 1, the source signals are independent; however, their signal mixtures are not.
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Curvelets have been used in place of the Gaussian filter and gradient estimation to compute a vector field whose directions and magnitudes approximate the direction and strength of edges in the image, to which steps 3 - 5 of the Canny algorithm are then applied. Curvelets decompose signals into separate components of different scales, and dropping the components of finer scales can reduce noise[12].
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Median filtering is one kind of smoothing technique, as is linear Gaussian filtering. All smoothing techniques are effective at removing noise in smooth patches or smooth regions of a signal, but adversely affect edges. Often though, at the same time as reducing the noise in a signal, it is important to preserve the edges. Edges are of critical importance to the visual appearance of images, for example.
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The Gaussian sphere has accumulator cells that increase when a great circle passes through them, i.e. in the image a line segment intersects the vanishing point. Several modifications have been made since, but one of the most efficient techniques was using the Hough Transform, mapping the parameters of the line segment to the bounded space. Cascaded Hough Transforms have been applied for multiple vanishing points.
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An interesting phenomenon related to the filament propagation is the refocusing of focused laser pulses after the geometrical focus.M. Mlejnek, E.M. Wright, J.V. Moloney, Opt. Lett. 23 1998 382A. Talebpour, S. Petit, S.L. Chin, Re-focusing during the propagation of a focused femtosecondTi:Sapphire laser pulse in air, Optics Communications 171 1999 285–290 Gaussian Beam propagation predicts an increasing beam width bidirectionally away from the geometric focus.
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This overall framework has been applied to a large variety of problems in computer vision, including feature detection, feature classification, image segmentation, image matching, motion estimation, computation of shape cues and object recognition. The set of Gaussian derivative operators up to a certain order is often referred to as the N-jet and constitutes a basic type of feature within the scale-space framework.
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In many practical applications, the true value of σ is unknown. As a result, we need to use a distribution that takes into account that spread of possible σ's. When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. The standard error is the standard deviation of the Student t-distribution.
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Finally A gets transformed by the private transformation S to give the valid signature x = S^{-1}(A) . The system of equations becomes linear if the vinegar variables are fixed – no oil variable is multiplied with another oil variable in the equation. Therefore, the oil variables can be computed easily using, for example, a Gaussian reduction algorithm. The signature creation is itself fast and computationally easy.
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All formulae in this article are correct in SI units; they may need to be changed for use in other unit systems. For example, in SI units, a loop of current with current and area has magnetic moment (see below), but in Gaussian units the magnetic moment is . Other units for measuring the magnetic dipole moment include the Bohr magneton and the nuclear magneton.
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Typical (single-replica) MTD simulations can include up to 3 CVs, even using the multi-replica approach, it is hard to exceed 8 CVs, in practice. This limitation comes from the bias potential, constructed by adding Gaussian functions (kernels). It is a special case of the kernel density estimator (KDE). The number of required kernels, for a constant KDE accuracy, increases exponentially with the number of dimensions.
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In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
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In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. It was invented by Austrian mathematician Bruno Buchberger. One can view it as a generalization of the Euclidean algorithm for univariate GCD computation and of Gaussian elimination for linear systems.
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This is a technique which trades quality for speed. Here, every volume element is splatted, as Lee Westover said, like a snow ball, on to the viewing surface in back to front order. These splats are rendered as disks whose properties (color and transparency) vary diametrically in normal (Gaussian) manner. Flat disks and those with other kinds of property distribution are also used depending on the application.
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Standard brain maps such as the Talairach-Tournoux or templates from the Montréal Neurological Institute (MNI) allow researchers from across the world to compare their results. Images can be smoothed to make the data less noisy (similar to the 'blur' effect used in some image-editing software) by which voxels are averaged with their neighbours, typically using a Gaussian filter or by wavelet transformation.
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In mathematical finance, the Cheyette Model is a quasi-Gaussian, quadratic volatility model of interest rates intended to overcome certain limitations of the Heath-Jarrow-Morton framework. By imposing a special time dependent structure on the forward rate volatility function, the Cheyette approach allows for dynamics which are Markovian, in contrast to the general HJM model. This in turn allows the application of standard econometric valuation concepts.
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In some cases, such as the air conditioner example, the distribution of survival times may be approximated well by a function such as the exponential distribution. Several distributions are commonly used in survival analysis, including the exponential, Weibull, gamma, normal, log-normal, and log-logistic. These distributions are defined by parameters. The normal (Gaussian) distribution, for example, is defined by the two parameters mean and standard deviation.
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Bearpark is a Principal Research Fellow in the Chemistry Department at Imperial College London. He works in computational chemistry, including method and software development with applications to modeling the excited electronic states of large molecules and their photochemical reaction dynamics, as well as research into the coherent control of chemical reactions. He has also contributed to the development of the Gaussian computational chemistry codes.
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The two chapters on special relativity were rewritten entirely, with the basic results of relativistic kinematics being moved to the problems and replaced by a discussion on the electromagnetic Lagrangian. Materials on transition and collision radiation and multipole fields were modified. 117 new problems were added. While the previous two editions use Gaussian units, the third uses SI units, albeit for the first ten chapters only.
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Bayesian Quadrature is a statistical approach to the numerical problem of computing integrals and falls under the field of probabilistic numerics. It can provide a full handling of the uncertainty over the solution of the integral expressed as a Gaussian Process posterior variance. It is also known to provide very fast convergence rates which can be up to exponential in the number of quadrature points n.
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London etc.: Prentice Hall, Page 326. Edge detectors that perform better than the Canny usually require longer computation times or a greater number of parameters. The Canny–Deriche detector was derived from similar mathematical criteria as the Canny edge detector, although starting from a discrete viewpoint and then leading to a set of recursive filters for image smoothing instead of exponential filters or Gaussian filters.
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The refractive index has two primary characteristics, the refractive index profile, and the offset. Typically, the refractive index profile can be uniform or apodized, and the refractive index offset is positive or zero. There are six common structures for FBGs; # uniform positive-only index change, # Gaussian apodized, # raised-cosine apodized, # chirped, # discrete phase shift, and # superstructure. The first complex grating was made by J. Canning in 1994.
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In the GPU CUDA implementation, each EMD, is mapped to a thread. The memory layout, especially of high-dimensional data, is rearranged to meet memory coalescing requirements and fit into the 128-byte cache lines. The data is first loaded along the lowest dimension and then consumed along a higher dimension. This step is performed when the Gaussian noise is added to form the ensemble data.
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Regarding the topic of automatic scale selection based on normalized derivatives, pyramid approximations are frequently used to obtain real-time performance.Crowley, J, Riff O: Fast computation of scale normalised Gaussian receptive fields, Proc. Scale- Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, 2003.Lowe, D. G., “Distinctive image features from scale- invariant keypoints”, International Journal of Computer Vision, 60, 2, pp.
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Using this grid, the function values are calculated at each grid point. To do this the method utilises a series of Gaussian functions, given a distance weighting in order to determine the relative importance of any given measurement on the determination of the function values. Correction passes are then made to optimise the function values, by accounting for the spectral response of the interpolated points.
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Data sets are placed within the 4-D space by use of a data descriptor file. GrADS interprets station data as well as gridded data, and the grids may be regular, non-linearly spaced, Gaussian, or of variable resolution. Data from different data sets may be graphically overlaid, with correct spatial and time registration. It uses the ctl mechanism to join differing time group data sets.
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Let U be the unit disk with Poincaré metric \rho; let S be a Riemann surface endowed with a Hermitian metric \sigma whose Gaussian curvature is ≤ −1; let f:U\rightarrow S be a holomorphic function. Then :\sigma(f(z_1),f(z_2)) \leq \rho(z_1,z_2) for all z_1,z_2 \in U. A generalization of this theorem was proved by Shing-Tung Yau in 1973.
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Thus far the theory only considers mean values of continuous distributions corresponding to an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M (the moment matrix of the Gaussian). And this may also affect the efficiency of the process. Unfortunately very little is known about this, at least theoretically.
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In these cases, peak-to-peak measurements may be more useful. Many efforts have been made to meaningfully quantify distributions that are neither Gaussian nor have a meaningful peak level. All have shortcomings but most tend to be good enough for the purposes of engineering work. In computer networking, jitter can refer to packet delay variation, the variation (statistical dispersion) in the delay of the packets.
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CP2K is a freely available (GPL) quantum chemistry and solid state physics program package, written in Fortran 2003, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It provides a general framework for different methods: density functional theory (DFT) using a mixed Gaussian and plane waves approach (GPW) via LDA, GGA, MP2, or RPA levels of theory, classical pair and many-body potentials, semi-empirical (AM1, PM3, MNDO, MNDOd, PM6) and tight-binding Hamiltonians, as well as Quantum Mechanics/Molecular Mechanics (QM/MM) hybrid schemes relying on the Gaussian Expansion of the Electrostatic Potential (GEEP). CP2K can do simulations of molecular dynamics, metadynamics, Monte Carlo, Ehrenfest dynamics, vibrational analysis, core level spectroscopy, energy minimization, and transition state optimization using NEB or dimer method. CP2K provides editor plugins for Vim and Emacs syntax highlighting, along with other tools for input generation and output processing.
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Triple correlation methods are frequently used in signal processing for treating signals that are corrupted by additive white Gaussian noise; in particular, triple correlation techniques are suitable when multiple observations of the signal are available and the signal may be translating in between the observations, e.g.,a sequence of images of an object translating on a noisy background. What makes the triple correlation particularly useful for such tasks are three properties: (1) it is invariant under translation of the underlying signal; (2) it is unbiased in additive Gaussian noise; and (3) it retains nearly all of the relevant phase information in the underlying signal. Properties (1)-(3) of the triple correlation extend in many cases to functions on an arbitrary locally compact group, in particular to the groups of rotations and rigid motions of euclidean space that arise in computer vision and signal processing.
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There are interesting relations between scale-space representation and biological vision and hearing. Neurophysiological studies of biological vision have shown that there are receptive field profiles in the mammalian retina and visual cortex, which can be well modelled by linear Gaussian derivative operators, in some cases also complemented by a non-isotropic affine scale-space model, a spatio- temporal scale-space model and/or non-linear combinations of such linear operators. Regarding biological hearing there are receptive field profiles in the inferior colliculus and the primary auditory cortex that can be well modelled by spectra-temporal receptive fields that can be well modelled by Gaussian derivates over logarithmic frequencies and windowed Fourier transforms over time with the window functions being temporal scale-space kernels.T. Lindeberg and A. Friberg "Idealized computational models of auditory receptive fields", PLOS ONE, 10(3): e0119032, pages 1–58, 2015T.
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Unusual waves have been studied scientifically for many years (for example, John Scott Russell's Wave of Translation, an 1834 study of a soliton wave), but these were not linked conceptually to sailors' stories of encounters with giant rogue ocean waves, as the latter were believed to be scientifically implausible. Since the 19th century, oceanographers, meteorologists, engineers and ship designers have used a statistical model known as the Gaussian function (or Gaussian Sea or standard linear model) to predict wave height, on the assumption that wave heights in any given sea are tightly grouped around a central (the average of the largest third) value, known as the 'significant wave height'. In a storm sea with a significant wave height of , the model suggests there will hardly ever be a wave higher than . It suggests one of could indeed happen – but only once in ten thousand years (of wave height of ).
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The error associated with the paraxial approximation. In this plot the cosine is approximated by . In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (θ) to the optical axis of the system, and lies close to the axis throughout the system.
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There many other risk measures (like coherent risk measures) might better reflect investors' true preferences. Modern portfolio theory has also been criticized because it assumes that returns follow a Gaussian distribution. Already in the 1960s, Benoit Mandelbrot and Eugene Fama showed the inadequacy of this assumption and proposed the use of more general stable distributions instead. Stefan Mittnik and Svetlozar Rachev presented strategies for deriving optimal portfolios in such settings.
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The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical isoperimetric inequality. Sergey Bobkov proved a functional generalization of the Gaussian isoperimetric inequality, from a certain "two point analytic inequality". Bakry and Ledoux gave another proof of Bobkov's functional inequality based on the semigroup techniques which works in a much more abstract setting. Later Barthe and Maurey gave yet another proof using the Brownian motion.
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Typically digital system will encode bits with uniform probability to maximize the entropy. Shaping code act as buffer between digital sources and modulator communication system. They will receive uniformly distributed data and convert it to Gaussian like distribution before presenting to the modulator. Shaping codes are helpful in reducing transmit power and thus reduce the cost of Power amplifier and the interference caused to other users in the vicinity.
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The Brunn–Minkowski inequality asserts that the Lebesgue measure is log- concave. The restriction of the Lebesgue measure to any convex set is also log-concave. By a theorem of Borell, a measure is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a logarithmically concave function. Thus, any Gaussian measure is log-concave.
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This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking. An intrinsic definition of the Gaussian curvature at a point is the following: imagine an ant which is tied to with a short thread of length . It runs around while the thread is completely stretched and measures the length of one complete trip around . If the surface were flat, the ant would find .
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Model B retains preferential attachment but eliminates growth. The model begins with a fixed number of disconnected nodes and adds links, preferentially choosing high degree nodes as link destinations. Though the degree distribution early in the simulation looks scale-free, the distribution is not stable, and it eventually becomes nearly Gaussian as the network nears saturation. So preferential attachment alone is not sufficient to produce a scale-free structure.
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Consider a simple non-hierarchical Bayesian model consisting of a set of i.i.d. observations from a Gaussian distribution, with unknown mean and variance.Based on Chapter 10 of Pattern Recognition and Machine Learning by Christopher M. Bishop In the following, we work through this model in great detail to illustrate the workings of the variational Bayes method. For mathematical convenience, in the following example we work in terms of the precision — i.e.
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The magnitudes are further weighted by a Gaussian function with \sigma equal to one half the width of the descriptor window. The descriptor then becomes a vector of all the values of these histograms. Since there are 4 × 4 = 16 histograms each with 8 bins the vector has 128 elements. This vector is then normalized to unit length in order to enhance invariance to affine changes in illumination.
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The worst-case computational complexity of Khachiyan's ellipsoidal algorithm is a polynomial. The criss-cross algorithm has exponential complexity. The time complexity of an algorithm counts the number of arithmetic operations sufficient for the algorithm to solve the problem. For example, Gaussian elimination requires on the order of D3 operations, and so it is said to have polynomial time-complexity, because its complexity is bounded by a cubic polynomial.
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Another perspective on concept granulation may be obtained from work on parametric models of categories. In mixture model learning, for example, a set of data is explained as a mixture of distinct Gaussian (or other) distributions. Thus, a large amount of data is "replaced" by a small number of distributions. The choice of the number of these distributions, and their size, can again be viewed as a problem of concept granulation.
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The Student-t distribution, the Irwin–Hall distribution and the Bates distribution also extend the normal distribution, and include in the limit the normal distribution. So there is no strong reason to prefer the "generalized" normal distribution of type 1, e.g. over a combination of Student-t and a normalized extended Irwin–Hall – this would include e.g. the triangular distribution (which cannot be modeled by the generalized Gaussian type 1).
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PUFF-PLUME is a model used to help predict how air pollution disperses in the atmosphere. It is a Gaussian atmospheric transport chemical/radionuclide dispersion model that includes wet and dry deposition, real-time input of meteorological observations and forecasts, dose estimates from inhalation and gamma shine (i.e., radiation), and puff or continuous plume dispersion modes. It was first developed by the Pacific Northwest National Laboratory (PNNL) in the 1970s.
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The astronomical unit of length is now defined as exactly 149 597 870 700 meters. It is approximately equal to the mean Earth–Sun distance. It was formerly defined as that length for which the Gaussian gravitational constant (k) takes the value when the units of measurement are the astronomical units of length, mass and time. The dimensions of k2 are those of the constant of gravitation (G), i.e.
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The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates.M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling. 2nd ed.
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Abmho or absiemens is a unit of electrical conductance in the centimetre gram second (emu-cgs) system of units. It's equal to gigasiemens (inverse of nano- ohm). The emu-cgs units are one of several systems of electromagnetic units within the centimetre gram second system of units; others include esu-cgs, Gaussian units, and Lorentz–Heaviside units. In these other systems, the abmho is not one of the units.
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Gaussian curve with a two-dimensional domain Many shapes have metaphorical names, i.e., their names are metaphors: these shapes are named after a most common object that has it. For example, "U-shape" is a shape that resembles the letter U, a bell-shaped curve has the shape of the vertical cross-section of a bell, etc. These terms may variously refer to objects, their cross sections or projections.
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In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that the subcomponents are non-Gaussian signals and that they are statistically independent from each other. ICA is a special case of blind source separation. A common example application is the "cocktail party problem" of listening in on one person's speech in a noisy room.
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In this section we will walk through the steps of a bottom-up document layout analysis algorithm developed in 1993 by O`Gorman. The steps in this approach are as follows: # Preprocess the image to remove Gaussian and salt-and-pepper noise. Note that some noise removal filters may consider commas and periods as noise, so some care must be taken. # Convert the image into a binary image, i.e.
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The multi-agent system Zhang mentioned could be used to describe an engineering or economic system. The uncertainty in his work is a kind of random noise appearing in the agent’s dynamic model. Brownian agent swarm systems are such examples, where the acceleration of agent depends on not only its own state variables (e.g. position, velocity, and energy), control, Gaussian white noise, but also the population position average.
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The diameter of the Riemannian circle is π, in contrast with the usual value of 2 for the Euclidean diameter of the unit circle. The inclusion of the Riemannian circle as the equator (or any great circle) of the 2-sphere of constant Gaussian curvature +1, is an isometric imbedding in the sense of metric spaces (there is no isometric imbedding of the Riemannian circle in Hilbert space in this sense).
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Other types of curves, such as trigonometric functions (such as sine and cosine), may also be used, in certain cases. In spectroscopy, data may be fitted with Gaussian, Lorentzian, Voigt and related functions. In agriculture the inverted logistic sigmoid function (S-curve) is used to describe the relation between crop yield and growth factors. The blue figure was made by a sigmoid regression of data measured in farm lands.
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3 (pages 200–206), Devroye (1986). The MSM package in R has a function, rtnorm, that calculates draws from a truncated normal. The truncnorm package in R also has functions to draw from a truncated normal. Chopin (2011) proposed (arXiv) an algorithm inspired from the Ziggurat algorithm of Marsaglia and Tsang (1984, 2000), which is usually considered as the fastest Gaussian sampler, and is also very close to Ahrens’s algorithm (1995).
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In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of integers, such as , and the complex number , which generates the Gaussian integers.
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Probabilistic mixture models such as Gaussian mixture models (GMM) are used to resolve point set registration problems in image processing and computer vision fields. For pair-wise point set registration, one point set is regarded as the centroids of mixture models, and the other point set is regarded as data points (observations). State-of-the-art methods are e.g. coherent point drift (CPD) and Student's t-distribution mixture models (TMM).
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As an alternative to the EM algorithm, the mixture model parameters can be deduced using posterior sampling as indicated by Bayes' theorem. This is still regarded as an incomplete data problem whereby membership of data points is the missing data. A two-step iterative procedure known as Gibbs sampling can be used. The previous example of a mixture of two Gaussian distributions can demonstrate how the method works.
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In probability theory, a branch of mathematics, white noise analysis , otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener process. It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes. The term white noise was first used for signals with a flat spectrum.
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A graph of the Gaussian function . The coloured region between the function and the x-axis has area . The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968, pp. 174–190.
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Phase truncation spurs can be reduced substantially by the introduction of white gaussian noise prior to truncation. The so-called dither noise is summed into the lower W+1 bits of the PA output word to linearize the truncation operation. Often the improvement can be achieved without penalty because the DAC noise floor tends to dominate system performance. Amplitude truncation spurs can not be mitigated in this fashion.
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The problem with the methods described above is that errors will accumulate and the series will tend to diverge from the true function. A solution which guarantees a constant maximum error is to use curve fitting. A minimum of N values are calculated evenly spaced along the range of the desired calculations. Using a curve fitting technique like Gaussian reduction an N−1th degree polynomial interpolation of the function is found.
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In 2007, Ravela et al. introduce the joint position-amplitude adjustment model using ensembles, and systematically derive a sequential approximation which can be applied to both EnKF and other formulations. Their method does not make the assumption that amplitudes and position errors are independent or jointly Gaussian, as others do. The morphing EnKF employs intermediate states, obtained by techniques borrowed from image registration and morphing, instead of linear combinations of states.
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One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients, e.g. Neville's algorithm. The cost is O(n2) operations, while Gaussian elimination costs O(n3) operations. Furthermore, you only need to do O(n) extra work if an extra point is added to the data set, while for the other methods, you have to redo the whole computation.
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In this section, we show achievability of the upper bound on the rate from the last section. A codebook, known to both encoder and decoder, is generated by selecting codewords of length n, i.i.d. Gaussian with variance P-\epsilon and mean zero. For large n, the empirical variance of the codebook will be very close to the variance of its distribution, thereby avoiding violation of the power constraint probabilistically.
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The Improved Layer 2 Protocol (IL2P) was created by Nino Carrillo, KK4HEJ, based on AX.25 and implements Reed Solomon Forward Error Correction for greater accuracy and throughput than either AX.25 or FX.25. Specifically, in order to achieve greater stability on links exceeding speeds of 1200 baud. IL2P can be used with a variety of modulation methods including Audio frequency shift keying and Gaussian Frequency Shift Keying.
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Some edge-detection operators are instead based upon second-order derivatives of the intensity. This essentially captures the rate of change in the intensity gradient. Thus, in the ideal continuous case, detection of zero- crossings in the second derivative captures local maxima in the gradient. The early Marr–Hildreth operator is based on the detection of zero-crossings of the Laplacian operator applied to a Gaussian-smoothed image.
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The sample has size 256 μm2, the distance between the lines is 17 μm, and the PSF is a two-dimensional Gaussian kernel with a FWHM of 35 μm. The BSIPAM experiment is repeated M = 200 times, each with random speckle patterns, and the speckles each have a size of 9 μm. Figure 7 displays the results. When BSIPAM is applied, the resolution is improved by a factor of 2.4.
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Given a manifold in three dimensions that is smooth and differentiable over a patch containing the point p, where k and m are defined as the principal curvatures and K(x) is the Gaussian curvature at a point x, if k has a max at p, m has a min at p, and k is strictly greater than m at p, then K(p) is a non-positive real number..
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The resulting image, with white Gaussian noise removed is shown below the original image. When filtering any form of data it is important to quantify the signal-to-noise-ratio of the result. In this case, the SNR of the noisy image in comparison to the original was 30.4958%, and the SNR of the denoised image is 32.5525%. The resulting improvement of the wavelet filtering is a SNR gain of 2.0567%.
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Herbert Sichel (1915–1995) was a statistician who made great advances in the areas of both theoretical and applied statistics. He developed the Sichel-t estimator for the log-normal distribution's t-statistic. He also made great leaps in the area of the generalized inverse Gaussian distribution which became known as the Sichel distribution. Dr Sichel pioneered the science of geostatistics with Danie Krige in the early 1950s.
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The near-horizon metric (NHM) refers to the near-horizon limit of the global metric of a black hole. NHMs play an important role in studying the geometry and topology of black holes, but are only well defined for extremal black holes. NHMs are expressed in Gaussian null coordinates, and one important property is that the dependence on the coordinate r is fixed in the near- horizon limit.
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Quasioptics concerns the propagation of electromagnetic radiation when the size of the wavelength is comparable to the size of the optical components (e.g. lenses, mirrors, and apertures) and hence diffraction effects become significant. It commonly describes the propagation of Gaussian beams where the beam width is comparable to the wavelength. This is in contrast to geometrical optics, where the wavelength is small compared to the relevant length scales.
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The result was first stated and proved by V. N. Sudakov, as pointed out in a paper by Dudley, "V. N. Sudakov's work on expected suprema of Gaussian processes," in High Dimensional Probability VII, Eds. C. Houdré, D. M. Mason, P. Reynaud-Bouret, and Jan Rosiński, Birkhăuser, Springer, Progress in Probability 71, 2016, pp. 37–43. Dudley had earlier credited Volker Strassen with making the connection between entropy and regularity.
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Developed by David Horn and Marvin Weinstein in 2009, Dynamic Quantum Clustering (DQC) approaches the complexity problem from a different heading than AQC. Using a mathematical shortcut to simplify the gradient descent, it also features the ability of nearby points in adjacent local minima to "tunnel" and resolve to a single cluster. A tunneling hyper-parameter determines whether or not a data point "tunnels" based on the width of the Gaussian.
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1 . Once this area is known, the area of a polygon may be computed by summing the contributions from all the edges of the polygon. Here an expression for the area of is developed following . The area of any closed region of the ellipsoid is : T = \int dT = \int \frac1K \cos\varphi\,d\varphi\,d\lambda, where is an element of surface area and is the Gaussian curvature.
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While systems of three or four equations can be readily solved by hand (see Cracovian), computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as pivoting.
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In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the expense of computing the Faddeeva function, the Voigt profile is often approximated using a pseudo-Voigt profile.
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In behavioral experiments, test individuals are usually presented with pairs of signals from different sensory modalities (such as visual and audio) at different SOAs and asked to make either synchrony judgements (i.e. if the pair of signals appears to have come at the exact same time) or temporal order judgements (i.e. which signal appears to have come earlier than the other). Results from an individual’s synchrony judgement tasks are typically fitted to a Gaussian curve with average perceived synchrony in percentage (between 0 and 1) on the y-axis and SOA (in milliseconds) on the x-axis, and the PSS of this individual is defined as the mean of the Gaussian distribution. Alternatively, results from an individual’s temporal order judgement tasks are typically fitted to an S-shaped logistic psychometric curve, with percentage of trials where the subject responds that signals from one certain modality has come first on the y-axis and SOA (in ms) on the x-axis.
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The least-squares approach implicitly assumes that the errors in the image data have a Gaussian distribution with zero mean. If one expects the window to contain a certain percentage of "outliers" (grossly wrong data values, that do not follow the "ordinary" Gaussian error distribution), one may use statistical analysis to detect them, and reduce their weight accordingly. The Lucas–Kanade method per se can be used only when the image flow vector V_x,V_y between the two frames is small enough for the differential equation of the optical flow to hold, which is often less than the pixel spacing. When the flow vector may exceed this limit, such as in stereo matching or warped document registration, the Lucas–Kanade method may still be used to refine some coarse estimate of the same, obtained by other means; for example, by extrapolating the flow vectors computed for previous frames, or by running the Lucas-Kanade algorithm on reduced-scale versions of the images.
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For one-dimensional kernels, there is a well-developed theory of multi-scale approaches, concerning filters that do not create new local extrema or new zero-crossings with increasing scales. For continuous signals, filters with real poles in the s-plane are within this class, while for discrete signals the above-described recursive and FIR filters satisfy these criteria. Combined with the strict requirement of a continuous semi-group structure, the continuous Gaussian and the discrete Gaussian constitute the unique choice for continuous and discrete signals. There are many other multi- scale signal processing, image processing and data compression techniques, using wavelets and a variety of other kernels, that do not exploit or require the same requirements as scale space descriptions do; that is, they do not depend on a coarser scale not generating a new extremum that was not present at a finer scale (in 1D) or non-enhancement of local extrema between adjacent scale levels (in any number of dimensions).
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As part of the Bayesian framework, the Gaussian process specifies the prior distribution that describes the prior beliefs about the properties of the function being modeled. These beliefs are updated after taking into account observational data by means of a likelihood function that relates the prior beliefs to the observations. Taken together, the prior and likelihood lead to an updated distribution called the posterior distribution that is customarily used for predicting test cases.
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Lévy subordination is used to construct new Lévy processes (for example variance gamma process and normal inverse Gaussian process). There is a large number of financial applications of processes constructed by Lévy subordination. An additive process built via additive subordination maintains the analytical tractability of a process built via Lévy subordination but it reflects better the time-inhomogeneus structure of market data. Additive subordination is applied to the commodity market and to VIX options.
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It includes the continuous beta spectrum and K-, L-, and M-lines due to internal conversion. Since the binding energy of the K electrons in 203Tl amounts to 85 keV, the K line has an energy of 279 - 85 = 194 keV. Because of lesser binding energies, the L- and M-lines have higher energies. Because of the finite energy resolution of the spectrometer, the "lines" have a Gaussian shape of finite width.
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Stein's method is a mathematical technique originally developed for approximating random variables such as Gaussian and Poisson variables, which has also been applied to point processes. Stein's method can be used to derive upper bounds on probability metrics, which give way to quantify how different two random mathematical objects vary stochastically.A. D. Barbour and T. C. Brown. Stein's method and point process approximation. Stochastic Processes and their Applications, 43(1):9–31, 1992.
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From a different perspective, the Kell factor defines the effective resolution of a discrete display device since the full resolution cannot be used without viewing experience degradation. The actual sampled resolution will depend on the spot size and intensity distribution. For electron gun scanning systems, the spot usually has a Gaussian intensity distribution. For CCDs, the distribution is somewhat rectangular, and is also affected by the sampling grid and inter-pixel spacing.
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The light intensity in the focal plane distributed as airy disk, and has circular symmetry. A 2-dimensional Gaussian function is a good approximation for airy disk. By fitting this function to the spot one can find the parameters x_0 and y_0 that are the coordinates of the center of the spot, and of the end-to-end vector. The second technique is to find the center of intensity,Blumberg, S., et al.
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This gaussian can be represented by means of an average magnitude Mv and a variance σ2. This distribution of globular cluster luminosities is called the Globular Cluster Luminosity Function (GCLF). (For the Milky Way, Mv = , σ = magnitudes.) The GCLF has also been used as a "standard candle" for measuring the distance to other galaxies, under the assumption that the globular clusters in remote galaxies follow the same principles as they do in the Milky Way.
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In elution mode, substances typically emerge from a column in narrow, Gaussian peaks. Wide separation of peaks, preferably to baseline, is desired for maximum purification. The speed at which any component of a mixture travels down the column in elution mode depends on many factors. But for two substances to travel at different speeds, and thereby be resolved, there must be substantial differences in some interaction between the biomolecules and the chromatography matrix.
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In other situations, the system of equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g., the 2D Poisson problem). Simplified forms of Gaussian elimination have been developed for these situations. The textbook Numerical Mathematics by Quarteroni, Sacco and Saleri, lists a modified version of the algorithm which avoids some of the divisions (using instead multiplications), which is beneficial on some computer architectures.
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Although there is no general analytical expression for α, its value has been derived numerically for many beam profiles. The lower limit is α ≈ 1.86225, which corresponds to Townes beams, whereas for a Gaussian beam α ≈ 1.8962. For air, n0 ≈ 1, n2 ≈ 4×10−23 m2/W for λ = 800 nm, and the critical power is Pcr ≈ 2.4 GW, corresponding to an energy of about 0.3 mJ for a pulse duration of 100 fs.
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The homogeneous broadened emission line will have a Lorentzian profile (i.e. will be best fitted by a Lorentzian function), while the inhomogeneously broadened emission will have a Gaussian profile. One or more phenomena may be present at the same time, but if one has a wider fluctuation, it will be the one responsible for the character of the broadening. These effects are not limited to laser systems, or even to optical spectroscopy.
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Plot of signal intensity versus time fitted with a Gaussian function. The string 6EQUJ5, commonly misinterpreted as a message encoded in the radio signal, represents in fact the signal's intensity variation over time, expressed in the particular measuring system adopted for the experiment. The signal itself appeared to be an unmodulated continuous wave, although any modulation with a period of less than 10 seconds or longer than 72 seconds would not have been detectable.
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Flowchart showing how the MAS5 algorithm by Agilent works. Factor Analysis for Robust Microarray Summarization (FARMS) is a model-based technique for summarizing array data at perfect match probe level. It is based on a factor analysis model for which a Bayesian maximum a posteriori method optimizes the model parameters under the assumption of Gaussian measurement noise. According to the Affycomp benchmark FARMS outperformed all other summarizations methods with respect to sensitivity and specificity.
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A variety of terms are used in the literature to describe these cells, including chromatically opposed or -opponent, spectrally opposed or -opponent, opponent colour, colour opponent, opponent response, and simply, opponent cell. The opponent color theory can be applied to computer vision and implemented as the Gaussian color model and the natural-vision-processing model.Barghout, Lauren. (2014). "Visual taxometric approach to image segmentation using fuzzy-spatial taxon cut yields contextually relevant regions".
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Fred Optical Engineering Software (FRED) is a commercial 3D CAD computer program for optical engineering used to simulate the propagation of light through optical systems. Fred can handle both incoherent and coherent light using Gaussian beam propagation. The program offers a high level of visualization using a WYSIWYG (What You See Is What You Get) parametric interface. According to the publisher, Photon Engineering, the name "Fred" is not an acronym, and does not mean anything.
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P. Twinanda et al. (2014), “Fisher Kernel Based Task Boundary Retrieval in Laparoscopic Database with Single Video Query” problems. The Fisher Vector (FV), a special, approximate, and improved case of the general Fisher kernel, is an image representation obtained by pooling local image features. The FV encoding stores the mean and the covariance deviation vectors per component k of the Gaussian-Mixture-Model (GMM) and each element of the local feature descriptors together.
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However, use of a nonlinear transformation requires caution. The influences of the data values will change, as will the error structure of the model and the interpretation of any inferential results. These may not be desired effects. On the other hand, depending on what the largest source of error is, a nonlinear transformation may distribute the errors in a Gaussian fashion, so the choice to perform a nonlinear transformation must be informed by modeling considerations.
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In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but the Gaussian and mexican-hat functions are common choices, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons, the weights are converging to local estimates.
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In engineering and practical areas, SI is nearly universal and has been for decades."CGS", in How Many? A Dictionary of Units of Measurement, by Russ Rowlett and the University of North Carolina at Chapel Hill In technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so.For example, one widely used graduate electromagnetism textbook is Classical Electrodynamics by J.D. Jackson.
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Gigi Masin (born October 24, 1955) is an Italian composer, ambient musician and producer from Venice. He is best known for his 1986 LP Wind and as a member of Gaussian Curve, a trio with Jonny Nash and Young Marco. A member of Italy's underground electronic scene, Masin pressed Wind privately and only released it at a series of small concerts in 1986. Most of the remaining copies were destroyed when Masin's house was flooded.
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A basic pseudospectral method for optimal control is based on the covector mapping principle. Other pseudospectral optimal control techniques, such as the Bellman pseudospectral method, rely on node-clustering at the initial time to produce optimal controls. The node clusterings occur at all Gaussian points. Moreover, their structure can be highly exploited to make them more computationally efficient, as ad-hoc scaling and Jacobian computation methods, involving dual number theory have been developed.
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A common application is public-key cryptography, whose algorithms commonly employ arithmetic with integers having hundreds of digits. recommends important RSA keys be 2048 bits (roughly 600 digits). Another is in situations where artificial limits and overflows would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining optimal or near-optimal values for coefficients needed in formulae, for example the that appears in Gaussian integration.
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Speaker recognition is a pattern recognition problem. The various technologies used to process and store voice prints include frequency estimation, hidden Markov models, Gaussian mixture models, pattern matching algorithms, neural networks, matrix representation, vector quantization and decision trees. For comparing utterances against voice prints, more basic methods like cosine similarity are traditionally used for their simplicity and performance. Some systems also use "anti-speaker" techniques such as cohort models and world models.
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Current generation gaming systems are able to render 3D graphics using floating point frame buffers, in order to produce HDR images. To produce the bloom effect, the HDRR images in the frame buffer are convolved with a convolution kernel in a post- processing step, before converting to RGB space. The convolution step usually requires the use of a large gaussian kernel that is not practical for realtime graphics, causing programmers to use approximation methods.
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In ACT, a chunk's activation decreases as a function of the time since the chunk was created and increases with the number of times the chunk has been retrieved from memory. Chunks can also receive activation from Gaussian noise, and from their similarity to other chunks. For example, if "chicken" is used as a retrieval cue, "canary" will receive activation by virtue of its similarity to the cue (i.e., both are birds, etc.).
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Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row.
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The Gaussian function is for x \in (-\infty,\infty) and would theoretically require an infinite window length. However, since it decays rapidly, it is often reasonable to truncate the filter window and implement the filter directly for narrow windows, in effect by using a simple rectangular window function. In other cases, the truncation may introduce significant errors. Better results can be achieved by instead using a different window function; see scale space implementation for details.
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COSMO has been implemented in a number of quantum chemistry or semi-empirical codes such as ADF, GAMESS-US, Gaussian, MOPAC, NWChem, TURBOMOLE, and Q-Chem. A COSMO version of the polarizable continuum model PCM has also been developed. Depending on the implementation, the details of the cavity construction and the used radii, the segments representing the molecule surface and the x value for the dielectric scaling function ƒ(ε) may vary.
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More precisely, given any integer-valued polynomials P1,..., Pk in one unknown m all with constant term 0, there are infinitely many integers x, m such that x + P1(m), ..., x + Pk(m) are simultaneously prime. The special case when the polynomials are m, 2m, ..., km implies the previous result that there are length k arithmetic progressions of primes. Tao proved an analogue of the Green–Tao theorem for the Gaussian primes.
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Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Problems were done on a counting board and included the use of negative numbers as well as fractions. The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last.
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Afterwards he worked as a bank officer. In the winter term 1924/25 he started to study again and studied mathematics and physics at the University of Breslau. He completed his doctoral thesis in Number theory with the title Die Reziprozitätsformel für Gaußsche Summen in reell quadratischen Zahlkörpern (The Reciprocity law for Gaussian Sums in real quadratic number fields) in 1931, directed by Hans Rademacher.Biographic details are taken from his CV in the doctoral thesis.
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Lauro Moscardini (born October 30, 1961) is an Italian astrophysicist and cosmologist. Moscardini has studied N-body cosmological simulations with non- Gaussian initial conditions. The research activity is mainly focussed in the field of theoretical and observational cosmology, in particular with the application of numerical techniques in astrophysics and the study of the formation of large cosmic structures. Moscardini's research is a mixture of observations and building models of large scale structures in the universe.
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Mixture models apply in the problem of directing multiple projectiles at a target (as in air, land, or sea defense applications), where the physical and/or statistical characteristics of the projectiles differ within the multiple projectiles. An example might be shots from multiple munitions types or shots from multiple locations directed at one target. The combination of projectile types may be characterized as a Gaussian mixture model.Spall, J. C. and Maryak, J. L. (1992).
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In the case of an infinite uniform (in z) cylindrically symmetric mass distribution we can conclude (by using a cylindrical Gaussian surface) that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance (from the axis), regardless of any masses at a larger distance. For example, inside an infinite uniform hollow cylinder, the field is zero.
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This is the condition that it should be a subfield of where is a squarefree odd number. This result was introduced by in his Zahlbericht and by . In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods. For example if we take a prime number , has a normal integral basis consisting of all the -th roots of unity other than .
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The amount of blurring required to accurately model subsurface scattering in skin is still under active research, but performing only a single blur poorly models the true effects. To emulate the wavelength dependent nature of diffusion, the samples used during the (Gaussian) blur can be weighted by channel. This is somewhat of an artistic process. For human skin, the broadest scattering is in red, then green, and blue has very little scattering.
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One example of phase synchronization of multiple oscillators can be seen in the behavior of Southeast Asian fireflies. At dusk, the flies begin to flash periodically with random phases and a gaussian distribution of native frequencies. As night falls, the flies, sensitive to one another's behavior, begin to synchronize their flashing. After some time all the fireflies within a given tree (or even larger area) will begin to flash simultaneously in a burst.
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Simpson's rule, which is based on a polynomial of order 2, is also a Newton–Cotes formula. Quadrature rules with equally spaced points have the very convenient property of nesting. The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used. If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas.
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In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(R) when R is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form.
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Trojan wavepacket evolution animation Classical simulation of the Trojan wavepacket on 1982 home ZX Spectrum microcomputer. The packet is approximated by the ensemble of points initially randomly localized within the peek of a Gaussian and moving according to the Newton equations. The ensemble stays localized. For the comparison the second simulation follows when the strength of the circularly polarized electric (rotating) field is equal to zero and the packet (points) fully spreads around the circle.
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Landsberg studied the theory of functions of two variables and also the theory of higher dimensional curves. In particular he studied the role of these curves in the calculus of variations and in mechanics. He worked with ideas related to those of Weierstrass, Riemann and Heinrich Weber on theta functions and Gaussian sums. His most important work, however was his contribution to the development of the theory of algebraic functions of a single variable.
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B3LYP) in a Cartesian- Gaussian LCAO basis. All algorithms are O(N) or O(N lg N) for non-metallic systems. Periodic boundary conditions in 1, 2 and 3 dimensions have been implemented through the Lorentz field (\Gamma-point), and an internal coordinate geometry optimizer allows full (atom+cell) relaxation using analytic derivatives. Effective core potentials for energies and forces have been implemented, but Effective Core Potential (ECP) lattice forces do not work yet.
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A physical implementation of a viterbi decoder will not yield an exact maximum-likelihood stream due to quantization of the input signal, branch and path metrics, and finite traceback length. Practical implementations do approach within 1dB of the ideal. The output of a Viterbi decoder, when decoding a message damaged by an additive gaussian channel, has errors grouped in error bursts. Stefan Host, Rolf Johannesson, Dmitrij K. Zigangirod, Kamil Sh. Zigangirod, and Viktor V. Zyablod.
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In May 2019 Ono published a joint paper (co-authored with Don Zagier and two former students) in the Proceedings of the National Academy of Sciences on the Riemann Hypothesis. Their work proves a large portion of the Jensen-Polya criterion for the Riemann Hypothesis. However, the Riemann Hypothesis remains unsolved. Their work also establishes the Gaussian Unitary Ensemble random matrix condition in derivative aspect for the derivatives of the Riemann Xi function.
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Turbomole was developed in 1987 and turned into a mature program system under the control of Reinhart Ahlrichs and his collaborators. Turbomole can perform a large-scale quantum chemical simulations of molecules, clusters, and later periodic solids. Gaussian basis sets are used in Turbomole. The functionality of the program concentrates extensively on the electronic structure methods with effective cost-performance characteristics such as density functional theory, second–order Møller-Plesset and coupled cluster theory.
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International Journal of Computer Vision 30 (2): pp 77--116. for an overview of the theoretical background. The Harris affine detector relies on the combination of corner points detected through Harris corner detection, multi-scale analysis through Gaussian scale space and affine normalization using an iterative affine shape adaptation algorithm. The recursive and iterative algorithm follows an iterative approach to detecting these regions: # Identify initial region points using scale-invariant Harris- Laplace Detector.
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Although 57 is not prime, it is jokingly known as the "Grothendieck prime" after a story in which mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number. This story is repeated in Part 2 of a biographical article on Grothendieck in Notices of the American Mathematical Society. As a semiprime, 57 is a Blum integer since its two prime factors are both Gaussian primes. 57 is a 20-gonal number.
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The grayscale value of each pixel can be used to provide sub-pixel accuracy by finding the centroid of the Gaussian. An object with markers attached at known positions is used to calibrate the cameras and obtain their positions and the lens distortion of each camera is measured. If two calibrated cameras see a marker, a three-dimensional fix can be obtained. Typically a system will consist of around 2 to 48 cameras.
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Note that Xt,p is a larger space than Xs,p, and in fact thee random variable u is almost surely not in the smaller space Xs,p. The space Xs,p is rather the Cameron-Martin space of this probability measure in the Gaussian case p = 2\. The random variable u is said to be Besov distributed with parameters (κ, s, p), and the induced probability measure is called a Besov measure.
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Kernel matrix defines the proximity of the input information. For example, in Gaussian Radial basis function, determines the dot product of the inputs in a higher-dimensional space, called feature space. It is believed that the data become more linearly separable in the feature space, and hence, linear algorithms can be applied on the data with a higher success. The kernel matrix can thus be analyzed in order to find the optimal number of clusters .
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The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a triangular pattern. In essence, the TNC decoding process, like the LNC decoding process involves Gaussian elimination. However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of triangularization,J. B. Fraleigh, and R. A. Beauregard, Linear Algebra.
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The atomic orbitals used are typically those of hydrogen-like atoms since these are known analytically i.e. Slater-type orbitals but other choices are possible such as the Gaussian functions from standard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials. By minimizing the total energy of the system, an appropriate set of coefficients of the linear combinations is determined. This quantitative approach is now known as the Hartree–Fock method.
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Spectral line shape describes the form of a feature, observed in spectroscopy, corresponding to an energy change in an atom, molecule or ion. Ideal line shapes include Lorentzian, Gaussian and Voigt functions, whose parameters are the line position, maximum height and half-width. Actual line shapes are determined principally by Doppler, collision and proximity broadening. For each system the half-width of the shape function varies with temperature, pressure (or concentration) and phase.
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Fundamentals of Stack Gas Dispersion is a book devoted to the fundamentals of air pollution dispersion modeling of continuous, buoyant pollution plumes from stationary point sources. The first edition was published in 1979. The current fourth edition was published in 2005. The subjects covered in the book include atmospheric turbulence and stability classes, buoyant plume rise, Gaussian dispersion calculations and modeling, time-averaged concentrations, wind velocity profiles, fumigations, trapped plumes and gas flare stack plumes.
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The Rachev ratio can be used in both ex-ante and ex-post analyses. The 5% ETL and 5% ETR of a non-Gaussian return distribution. Although the most probable return is positive, the Rachev ratio is 0.7 < 1, which means that the excess loss is not balanced by the excess profit in the investment. In the ex-post analysis, the Rachev ratio is computed by dividing the corresponding two sample AVaR's.
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Wavelet functions are used for both time and frequency localisation. For example, one of the windows used in calculating the Fourier coefficients is the Gaussian window which is optimally concentrated in time and frequency. This optimal nature can be explained by considering the time scaling and time shifting parameters a and b respectively. By choosing the appropriate values of a and b, we can determine the frequencies and the time associated with that signal.
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A steady state amplitude remains constant during time, thus is represented by a scalar. Otherwise, the amplitude is transient and must be represented as either a continuous function or a discrete vector. For audio, transient amplitude envelopes model signals better because many common sounds have a transient loudness attack, decay, sustain, and release. Other parameters can be assigned steady state or transient amplitude envelopes: high/low frequency/amplitude modulation, Gaussian noise, overtones, etc.
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In the context of the capacity of the narrowband Gaussian two-user multiple-access channel, CoverT. M. Cover, "Some advances in broadcast channels," in Advances in Communication Systems, A. Viterbi, Ed. New York: Academic Press, 1975, vol. 4, pp. 229-260 showed the achievability of the capacity region by means of a successive cancellation receiver, which decodes one user treating the other as noise, re- encodes its signal and subtracts it from the received signal.
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To describe the region around the point, a square region is extracted, centered on the interest point and oriented along the orientation as selected above. The size of this window is 20s. The interest region is split into smaller 4x4 square sub-regions, and for each one, the Haar wavelet responses are extracted at 5x5 regularly spaced sample points. The responses are weighted with a Gaussian (to offer more robustness for deformations, noise and translation).
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In practice it will suffice to update W only : W(i + 1) = (1 – b)W(i) + bygT. This is the formula used in a simple 2-dimensional model of a brain satisfying the Hebbian rule of associative learning; see the next section (Kjellström, 1996 and 1999). The figure below illustrates the effect of increased average information in a Gaussian p.d.f. used to climb a mountain Crest (the two lines represent the contour line).
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A simplified method of calculating chromatogram resolution is to use the plate model. The plate model assumes that the column can be divided into a certain number of sections, or plates and the mass balance can be calculated for each individual plate. This approach approximates a typical chromatogram curve as a Gaussian distribution curve. By doing this, the curve width is estimated as 4 times the standard deviation of the curve, 4σ.
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A mesh of a cactus showing the Gaussian Curvature at each vertex, using the angle defect method. Geometry processing involves working with a shape, usually in 2D or 3D, although the shape can live in a space of arbitrary dimensions. The processing of a shape involves three stages, which is known as its life cycle. At its "birth," a shape can be instantiated through one of three methods: a model, a mathematical representation, or a scan.
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In optics, a tophat (or top-hat) beam such as a laser beam or electron beam has a near-uniform fluence (energy density) within a circular disk. It is typically formed by diffractive optical elements from a Gaussian beam. Tophat beams are often used in industry, for example for laser drilling of holes in printed circuit boards. They are also used in very high power laser systems, which use chains of optical amplifiers to produce an intense beam.
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The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression, or shear.
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Visualisation of the Box–Muller transform — the coloured points in the unit square (u1, u2), drawn as circles, are mapped to a 2D Gaussian (z0, z1), drawn as crosses. The plots at the margins are the probability distribution functions of z0 and z1. Note that z0 and z1 are unbounded; they appear to be in [-2.5,2.5] due to the choice of the illustrated points. In the SVG file, hover over a point to highlight it and its corresponding point.
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They differ in the type of probability distribution and the gradient approximation method used. Different search spaces require different search distributions; for example, in low dimensionality it can be highly beneficial to model the full covariance matrix. In high dimensions, on the other hand, a more scalable alternative is to limit the covariance to the diagonal only. In addition, highly multi-modal search spaces may benefit from more heavy-tailed distributions (such as Cauchy, as opposed to the Gaussian).
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In this case, simple explicit formulae can be given for parameters of an imaging system such as focal length, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements. Gaussian optics is named after mathematician and physicist Carl Friedrich Gauss, who showed that an optical system can be characterized by a series of cardinal points, which allow one to calculate its optical properties.W.J. Smith, Modern Optical Engineering, 2007, McGraw-Hill, p. 22.
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A nonlinear Kalman filter which shows promise as an improvement over the EKF is the unscented Kalman filter (UKF). In the UKF, the probability density is approximated by a deterministic sampling of points which represent the underlying distribution as a Gaussian. The nonlinear transformation of these points are intended to be an estimation of the posterior distribution, the moments of which can then be derived from the transformed samples. The transformation is known as the unscented transform.
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In robotics, EKF SLAM is a class of algorithms which utilizes the extended Kalman filter (EKF) for simultaneous localization and mapping (SLAM). Typically, EKF SLAM algorithms are feature based, and use the maximum likelihood algorithm for data association. In the 1990s and 2000s, EKF SLAM had been the de facto method for SLAM, until the introduction of FastSLAM. Associated with the EKF is the gaussian noise assumption, which significantly impairs EKF SLAM's ability to deal with uncertainty.
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The method of Gaussian elimination appears - albeit without proof - in the Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 AD, but parts of it were written as early as approximately 150 BC., pp. 234–236 It was commented on by Liu Hui in the 3rd century.
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Cambridge University eventually published the notes as Arithmetica Universalis in 1707 long after Newton had left academic life. 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. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least- squares problems., p.
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McNicholas started his faculty career at the University of Guelph in 2007, and, in 2014, he moved to McMaster University. He has authored more than 100 scientific works and has been cited over 4000 times. The majority of his work is in the area of model-based clustering, specifically in developing novel finite mixture models for clustering and classification of multivariate data. He has published works on clustering high-dimensional data and the use of non-Gaussian mixtures.
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To discuss the modeling of unresolved scales, first the unresolved scales must be classified. They fall into two groups: resolved sub-filter scales (SFS), and sub-grid scales(SGS). The resolved sub-filter scales represent the scales with wave numbers larger than the cutoff wave number k_c, but whose effects are dampened by the filter. Resolved sub-filter scales only exist when filters non-local in wave-space are used (such as a box or Gaussian filter).
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These anomalous events have been shown to follow heavy-tailed statistics, also known as L-shaped statistics, fat-tailed statistics, or extreme-value statistics. These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition. Such distributions also describe the probabilities of freak ocean waves and various phenomena in both the man-made and natural worlds. Despite their infrequency, rare events wield significant influence in many systems.
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Parity learning is a problem in machine learning. An algorithm that solves this problem must find a function ƒ, given some samples (x, ƒ(x)) and the assurance that ƒ computes the parity of bits at some fixed locations. The samples are generated using some distribution over the input. The problem is easy to solve using Gaussian elimination provided that a sufficient number of samples (from a distribution which is not too skewed) are provided to the algorithm.
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Through this correspondence, more accuracy is obtained, and a statistical assessment of the results becomes possible. In this case, the calculation is adjusted with the Gaussian least squares method. A numerical value for the accuracy of the transformation parameters is obtained by calculating the values at the reference points, and weighting the results relative to the centroid of the points. While the method is mathematically rigorous, it is entirely dependent on the accuracy of the parameters that are used.
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The Gaussian beam photographic paper burn comparison of a carbon dioxide transversely-excited atmospheric-pressure laser, obtained during the optimization process by adjusting the alignment mirrors. The optical resonator, or optical cavity, in its simplest form is two parallel mirrors placed around the gain medium, which provide feedback of the light. The mirrors are given optical coatings which determine their reflective properties. Typically, one will be a high reflector, and the other will be a partial reflector.
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A general kind of lossy compression is to lower the resolution of an image, as in image scaling, particularly decimation. One may also remove less "lower information" parts of an image, such as by seam carving. Many media transforms, such as Gaussian blur, are, like lossy compression, irreversible: the original signal cannot be reconstructed from the transformed signal. However, in general these will have the same size as the original, and are not a form of compression.
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Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices. Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of indices.
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The linear least squares problem is to find the that minimizes , which is equivalent to projecting to the subspace spanned by the columns of . Assuming the columns of (and hence ) are independent, the projection solution is found from . Now is square () and invertible, and also equal to . But the lower rows of zeros in are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition).
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A microwave-range model of a quantum radar was proposed in 2015 by an international team and is based on the protocol of Gaussian quantum illumination. The basic concept is to create a stream of entangled visible- frequency photons and split it in half. One half, the "signal beam", goes through a conversion to microwave frequencies in a way that preserves the original quantum state. The microwave signal is then sent and received as in a normal radar system.
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The analytical solution is allowed for a very limited number of theoretical cases. Vice versa a large variety of instances may be quickly solved in an approximate way via the central limit theorem in terms of confidence interval around a Gaussian distribution – that's the benefit. The drawback is that the central limit theorem is applicable when the sample size is sufficiently large. Therefore, it is less and less applicable with the sample involved in modern inference instances.
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Examples of this work are expectation optimization of L2 f-divergence for stochastic variational Bayes inference, Gaussianized bridge sampling for Bayesian evidence, and BayesFast, a surrogate model based Hamiltonian Monte Carlo sampler. Seljak is developing machine learning methods with applications to cosmology, astronomy, and other sciences. Examples are Fourier based Gaussian process for analysis of time and/or spatially ordered data, generative models with explicit physics symmetries (translation, rotation), and sliced iterative transport methods for density estimation and sampling.
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Finding the same singular physical point in the two left and right images is known as the correspondence problem. Correctly locating the point gives the computer the capability to calculate the distance that the robot or camera is from the object. On the BH2 Lunar Rover the cameras use five steps: a bayer array filter, photometric consistency dense matching algorithm, a Laplace of Gaussian (LoG) edge detection algorithm, a stereo matching algorithm and finally uniqueness constraint.
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After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution over the objective function. The posterior distribution, in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines the next query point. There are several methods used to define the prior/posterior distribution over the objective function. The most common two methods use Gaussian Processes in a method called Kriging.
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In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
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For example, a Possion process will be Possion distributed at all points in time, or a Brownian motion will be normal distributed at all points in time. However, a Lévy process that is generalised hyperbolic at one point in time might fail to be generalized hyperbolic at another point in time. In fact, the generalized Laplace distributions and the normal inverse Gaussian distributions are the only subclasses of the generalized hyperbolic distributions that are closed under convolution.
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A radial cross- section through the Airy pattern (solid curve) and its Gaussian profile approximation (dashed curve). The abscissa is given in units of the wavelength \lambda times the f-number of the optical system. The Airy pattern falls rather slowly to zero with increasing distance from the center, with the outer rings containing a significant portion of the integrated intensity of the pattern. As a result, the root mean square (RMS) spotsize is undefined (i.e. infinite).
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In particular, financial crises are characterized by a significant increase in correlation of stock price movements which may seriously degrade the benefits of diversification. In a mean-variance optimization framework, accurate estimation of the variance-covariance matrix is paramount. Quantitative techniques that use Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective. Allowing the modeling process to allow for empirical characteristics in stock returns such as autoregression, asymmetric volatility, skewness, and kurtosis is important.
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The fast elliptic solver is based on fast Fourier analysis in both horizontal directions and Gaussian elimination in the vertical direction (Moussiopoulos and Flassak, 1989). The second deviation from the explicit treatment is related to the turbulent diffusion in vertical direction. In case of an explicit treatment of this term, the stability requirement may necessitate an unacceptable abridgement of the time increment. To avoid this, vertical turbulent diffusion is treated using the second order Crank–Nicolson method.
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The Process of Canny edge detection algorithm can be broken down to 5 different steps: # Apply Gaussian filter to smooth the image in order to remove the noise # Find the intensity gradients of the image # Apply non-maximum suppression to get rid of spurious response to edge detection # Apply double threshold to determine potential edges # Track edge by hysteresis: Finalize the detection of edges by suppressing all the other edges that are weak and not connected to strong edges.
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Minimum-shift keying (MSK) is another name for CPM with an excess bandwidth of 1/2 and a linear phase trajectory. Although this linear phase trajectory is continuous, it is not smooth since the derivative of the phase is not continuous. The spectral efficiency of CPM can be further improved by using a smooth phase trajectory. This is typically accomplished by filtering the phase trajectory prior to modulation, commonly using a raised cosine or a Gaussian filter.
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Consider a vector y formed by taking N observations of a fixed but unknown scalar parameter x disturbed by white Gaussian noise. We can describe the process by a linear equation y = 1x+ z, where 1 = [1,1,\ldots,1]^T. Depending on context it will be clear if 1 represents a scalar or a vector. Suppose that we know [-x_0,x_0] to be the range within which the value of x is going to fall in.
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As before, initial guesses of the parameters for the mixture model are made. Instead of computing partial memberships for each elemental distribution, a membership value for each data point is drawn from a Bernoulli distribution (that is, it will be assigned to either the first or the second Gaussian). The Bernoulli parameter θ is determined for each data point on the basis of one of the constituent distributions. Draws from the distribution generate membership associations for each data point.
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For most engineering applications, MKS (rationalized) or SI (Système International) units are commonly used. Two other sets of units, Gaussian and CGS-EMU, are the same for magnetic properties and are commonly used in physics. In all units, it is convenient to employ two types of magnetic field, B and H, as well as the magnetization M, defined as the magnetic moment per unit volume. # The magnetic induction field B is given in SI units of teslas (T).
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The variational Bayesian methods used for model estimation in DCM are based on the Laplace assumption, which treats the posterior over parameters as Gaussian. This approximation can fail in the context of highly non-linear models, where local minima may preclude the free energy from serving as a tight bound on log model evidence. Sampling approaches provide the gold standard; however, they are time consuming and have typically been used to validate the variational approximations in DCM.
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By utilizing a Gaussian Mean Shift Key modulation, communications with the satellite are achieved at 68.4 kbit/s or higher data rate. The satellite also uses open source software based on the Linux operating system. MidSTAR-1 has no attitude control or determination, no active thermal control, and its mass is 120 kg. One hundred percent success would be the successful launch and operation of the satellite with full support for the two primary experiments for two years.
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Histogram equalization is a non-linear transform which maintains pixel rank and is capable of normalizing for any monotonically increasing color transform function. It is considered to be a more powerful normalization transformation than the grey world method. The results of histogram equalization tend to have an exaggerated blue channel and look unnatural, due to the fact that in most images the distribution of the pixel values is usually more similar to a Gaussian distribution, rather than uniform.
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Wagon won the Lester R. Ford Award of the Mathematical Association of America for his 1988 paper, "Fourteen Proofs of a Result about Tiling a Rectangle".MAA Writing Awards: Fourteen Proofs of a Result about Tiling a Rectangle, retrieved 2012-03-10. Wagon and his co-authors Ellen Gethner and Brian Wick won the Chauvenet Prize for mathematical exposition in 2002 for their 1998 paper, "A Stroll through the Gaussian Primes".Chauvenet Prize, MAA, retrieved 2012-03-10.
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The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients.
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A value of 0 indicates no structural similarity. For an image, it is typically calculated using a sliding Gaussian window of size 11x11 or a block window of size 8×8. The window can be displaced pixel-by-pixel on the image to create an SSIM quality map of the image. In the case of video quality assessment, the authors propose to use only a subgroup of the possible windows to reduce the complexity of the calculation.
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When the Christoffel symbols are considered as being defined by the first fundamental form, the Gauss and Codazzi equations represent certain constraints between the first and second fundamental forms. The Gauss equation is particularly noteworthy, as it shows that the Gaussian curvature can be computed directly from the first fundamental form, without the need for any other information; equivalently, this says that can actually be written as a function of , even though the individual components cannot.
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The eigenvalues of are just the principal curvatures and at . In particular the determinant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shape operator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically.
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As an example, the Gaussian function is integrated from 0 to 1, i.e. the error function erf(1) ≈ 0.842700792949715. The triangular array is calculated row by row and calculation is terminated if the two last entries in the last row differ less than 10−8. 0.77174333 0.82526296 0.84310283 0.83836778 0.84273605 0.84271160 0.84161922 0.84270304 0.84270083 0.84270066 0.84243051 0.84270093 0.84270079 0.84270079 0.84270079 The result in the lower right corner of the triangular array is accurate to the digits shown.
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Jiayang Sun is an American statistician whose research has included work on simultaneous confidence bands for multiple comparisons, selection bias, mixture models, Gaussian random fields, machine learning, big data, statistical computing, graphics, and applications in biostatistics, biomedical research, software bug tracking, astronomy, and intellectual property law. She is a statistics professor, Bernard J. Dunn Eminent Scholar, and chair of the statistics department at George Mason University, and a former president of the Caucus for Women in Statistics.
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A surface with a parabolic line and its Gauss map. A ridge passes through the parabolic line giving rise to a cusp on the Gauss map. The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along a parabolic line) the Gauss map will have a fold catastrophe. This fold may contain cusps and these cusps were studied in depth by Thomas Banchoff, Terence Gaffney and Clint McCrory.
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Being a relatively new color space and having very specific uses, TSL hasn’t been widely implemented. Again, it is only very useful in skin detection algorithms. Skin detection itself can be used for a variety of applications – face detection, person tracking (for surveillance and cinematographic purposes), and pornography filtering are a few examples. A Self-Organizing Map (SOM) was implemented in skin detection using TSL and achieved comparable results to older methods of histograms and Gaussian mixture models.
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In elution mode, substances typically emerge from a column in narrow, Gaussian peaks. Wide separation of peaks, preferably to baseline, is desired in order to achieve maximum purification. The speed at which any component of a mixture travels down the column in elution mode depends on many factors. But for two substances to travel at different speeds, and thereby be resolved, there must be substantial differences in some interaction between the biomolecules and the chromatography matrix.
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Bayesian neural networks are a particular type of Bayesian network that results from treating deep learning and artificial neural network models probabilistically, and assigning a prior distribution to their parameters. Computation in artificial neural networks is usually organized into sequential layers of artificial neurons. The number of neurons in a layer is called the layer width. As layer width grows large, many Bayesian neural networks reduce to a Gaussian process with a closed form compositional kernel.
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Instead of dividing by n we can also use to create a similar distribution with a constant variance (like unity). By subtracting the mean we can set the resulting mean to zero. This way the parameter n would become a purely shape-adjusting parameter, and we obtain a distribution which covers the uniform, the triangular and, in the limit, also the normal Gaussian distribution. By allowing also non-integer n a highly flexible distribution can be created (e.g.
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Anisotropic diffusion can be used to remove noise from digital images without blurring edges. With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the heat equation which is equivalent to Gaussian blurring. This is ideal for removing noise but also indiscriminately blurs edges too. When the diffusion coefficient is chosen as an edge seeking function, such as in Perona-Malik, the resulting equations encourage diffusion (hence smoothing) within regions and prohibit it across strong edges.
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Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic of the torus is 0, since the complete graph on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a useful topological invariant, which has been extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.
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Daniel Leonard Ocone (born 1953) is a Professor in the Mathematics Department at Rutgers University, where he specializes in probability theory and stochastic processes.Source: Rutgers University web site He obtained his Ph.D at MIT in 1980 under the supervision of Sanjoy K. Mitter. He is known for the Clark–Ocone theorem in stochastic analysis. The continuous Ocone martingale is also named after him; it is a continuous martingale that is conditionally Gaussian, given its quadratic variation process.
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North American football The ball in North American football has a shape resembling a geometric lemon. However, although used with a related meaning in geometry, the term "football" is more commonly used to refer to a surface of revolution whose Gaussian curvature is positive and constant, formed from a more complicated curve than a circular arc. Alternatively, a football may refer to a more abstract orbifold, a surface modeled locally on a sphere except at two points.
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Fraunhofer diffraction returns then to be an asymptotic case that applies only when the input/output propagation distance is large enough to consider the quadratic phase term, within the Fresnel diffraction integral, negligible irrespectively to the actual curvature of the wavefront at the observation point. As the figures explain, the Gaussian pilot beam criterion allows describing the diffractive propagation for all the near/far field approximation cases set by the coarse criterion based on Fresnel number.
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This means that if we build a histogram of the realisations of the sum of independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as approaches infinity, this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
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Royen published this proof in an article with the title A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions on arXiv Supplemented by and subsequently in the Far East Journal of Theoretical Statistics,Thomas Royen: A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, in: Far East Journal of Theoretical Statistics, Part 48 Nr. 2, Pushpa Publishing House, Allahabad 2014, p.139–145 a relatively unknown periodical based in Allahabad, India, for which Royen was at the time voluntarily working as a referee himself. Due to this, his proof went at first largely unnoticed by the scientific community,In the Quanta magazine article for instance Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved. until in late 2015 two Polish mathematicians, Rafał Latała and Dariusz Matlak, wrote a paper in which they reorganized Royen's proof in a way that was intended to be easier to follow.
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T. Lindeberg and J. Garding "Shape- adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D structure". Image and Vision Computing 15 (6): pp 415–434, 1997. Hence, besides the commonly used multi-scale Harris operator, affine shape adaptation can be applied to other corner detectors as listed in this article as well as to differential blob detectors such as the Laplacian/difference of Gaussian operator, the determinant of the Hessian and the Hessian–Laplace operator.
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For example, if some mechanism allows the full transmission of the leading part of a pulse while strongly attenuating the pulse maximum and everything behind (distortion), the pulse maximum is effectively shifted forward in time, while the information on the pulse does not come faster than c without this effect. However, group velocity can exceed c in some parts of a Gaussian beam in vacuum (without attenuation). The diffraction causes the peak of the pulse to propagate faster, while overall power does not.
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Bayesian neural network with two hidden layers, transforming a 3-dimensional input (bottom) into a two-dimensional output (y_1, y_2) (top). Right: output probability density function p(y_1, y_2) induced by the random weights of the network. Video: as the width of the network increases, the output distribution simplifies, ultimately converging to a Neural network Gaussian process in the infinite width limit. Artificial neural networks are a class of models used in machine learning, and inspired by biological neural networks.
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The general procedure is as follows: the parameterized search distribution is used to produce a batch of search points, and the fitness function is evaluated at each such point. The distribution’s parameters (which include strategy parameters) allow the algorithm to adaptively capture the (local) structure of the fitness function. For example, in the case of a Gaussian distribution, this comprises the mean and the covariance matrix. From the samples, NES estimates a search gradient on the parameters towards higher expected fitness.
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Fernandes and Oliveira suggested an improved voting scheme for the Hough transform that allows a software implementation to achieve real-time performance even on relatively large images (e.g., 1280×960). The Kernel-based Hough transform uses the same (r,\theta) parameterization proposed by Duda and Hart but operates on clusters of approximately collinear pixels. For each cluster, votes are cast using an oriented elliptical-Gaussian kernel that models the uncertainty associated with the best-fitting line with respect to the corresponding cluster.
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These beams, made using axicons, provide an ideal optical trap to channel cold atoms. An article published by the research team at St. Andrews University in the UK in the Sept. 12 issue of Nature describes axicon use in optical tweezers, which are commonly used for manipulating microscopic particles such as cells and colloids. The tweezers use lasers with a Bessel beam profile produced by illuminating an axicon with a Gaussian beam, which can trap several particles along the beam's axis.
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Every subfield of a cyclotomic field is an abelian extension of the rationals. It follows that every nth root of unity may be expressed in term of k-roots, with various k not exceeding φ(n). In these cases Galois theory can be written out explicitly in terms of Gaussian periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.The Disquisitiones was published in 1801, Galois was born in 1811, died in 1832, but wasn't published until 1846.
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Non-Gaussian statistics arise due to the nonlinear mapping of random initial conditions into output states. For example, modulation instability amplifies input noise, which ultimately leads to soliton formation. Also, in systems displaying heavy- tailed statistical properties, random input conditions often enter through a seemingly insignificant, nontrivial, or otherwise-hidden variable. Such is generally the case for optical rogue waves; for example, they can begin from a specific out-of-band noise component, which is usually very weak and unnoticed.
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As described above, a Rayleigh fading channel itself can be modelled by generating the real and imaginary parts of a complex number according to independent normal Gaussian variables. However, it is sometimes the case that it is simply the amplitude fluctuations that are of interest (such as in the figure shown above). There are two main approaches to this. In both cases, the aim is to produce a signal that has the Doppler power spectrum given above and the equivalent autocorrelation properties.
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In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem. Another example are Gaussian integers, that is, numbers of the form , where and are integers, which can be used to classify sums of squares.
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In contrast, the statistical properties were found to be approximately Gaussian for low filament numbers. It was noted that extreme spatio-temporal events are found only in certain nonlinear media even though other media have larger nonlinear responses, and the experimental findings suggested that laser-induced thermodynamic fluctuations within the nonlinear medium are the origin of the extreme events observed in multifilamention. Numerical predictions of extreme occurrences in multiple beam filamentation have also been performed, with some differences in conditions and interpretation.
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First, for each candidate keypoint, interpolation of nearby data is used to accurately determine its position. The initial approach was to just locate each keypoint at the location and scale of the candidate keypoint. The new approach calculates the interpolated location of the extremum, which substantially improves matching and stability. The interpolation is done using the quadratic Taylor expansion of the Difference- of-Gaussian scale-space function, D \left( x, y, \sigma \right) with the candidate keypoint as the origin.
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Not all radiant energy is absorbed and turned into heat for welding. Some of the radiant energy is absorbed in the plasma created by vaporizing and then subsequently ionizing the gas. In addition, the absorptivity is affected by the wavelength of the beam, the surface composition of the material being welded, the angle of incidence, and the temperature of the material. Rosenthal point source assumption leaves an infinitely high temperature discontinuity which is addressed by assuming a Gaussian distribution instead.
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The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. cited in Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995.
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One way to help to conceptualize this is to consider a simple smoothing matrix like a Gaussian blur, used to mitigate data noise. In contrast to a simple linear or polynomial fit, computing the effective degrees of freedom of the smoothing function is not straight- forward. In these cases, it is important to estimate the Degrees of Freedom permitted by the H matrix so that the residual degrees of freedom can then be used to estimate statistical tests such as \chi^2 .
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Quite often the site energies feature a Gaussian distribution. Also the hopping distances can vary statistically (positional disorder). A consequence of the energetic broadening of the density of states (DOS) distribution is that charge motion is both temperature and field dependent and the charge carrier mobility can be several orders of magnitude lower than in an equivalent crystalline system. This disorder effect on charge carrier motion is diminished in organic field-effect transistors because current flow is confined in a thin layer.
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This hypothesis also provides for an alternative paradigm to explain power law manifestations that have been attributed to self-organized criticality. There are various mathematical models to create pink noise. Although self-organised criticality has been able to reproduce pink noise in sandpile models, these do not have a Gaussian distribution or other expected statistical qualities. It can be generated on computer, for example, by filtering white noise, inverse Fourier transform, or by multirate variants on standard white noise generation.
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The peak shape is usually a Gaussian distribution. In most spectra the horizontal position of the peak is determined by the gamma ray's energy, and the area of the peak is determined by the intensity of the gamma ray and the efficiency of the detector. The most common figure used to express detector resolution is full width at half maximum (FWHM). This is the width of the gamma ray peak at half of the highest point on the peak distribution.
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The Hubbert curve is an approximation of the production rate of a resource over time. It is a symmetric logistic distribution curve, often confused with the "normal" gaussian function. It first appeared in "Nuclear Energy and the Fossil Fuels," geologist M. King Hubbert's 1956 presentation to the American Petroleum Institute, as an idealized symmetric curve, during his tenure at the Shell Oil Company. It has gained a high degree of popularity in the scientific community for predicting the depletion of various natural resources.
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Discrete Laplace operator is often used in image processing e.g. in edge detection and motion estimation applications. The discrete Laplacian is defined as the sum of the second derivatives Laplace operator#Coordinate expressions and calculated as sum of differences over the nearest neighbours of the central pixel. Since derivative filters are often sensitive to noise in an image, the Laplace operator is often preceded by a smoothing filter (such as a Gaussian filter) in order to remove the noise before calculating the derivative.
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In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. Often the aim of reduction is to transform a matrix into its "row-reduced echelon form" or "row-echelon form"; this is the goal of Gaussian elimination.
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The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane The curvature of the normal section is called the normal curvature. If the surface is bow or cylinder shaped the maximum and the minimum of these curvatures are the principal curvatures. If the surface is saddle shaped the maxima of both sides are the principal curvatures. The product of the principal curvatures is the Gaussian curvature of the surface.
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The design of computer experiments has considerable differences from design of experiments for parametric models. Since a Gaussian process prior has an infinite dimensional representation, the concepts of A and D criteria (see Optimal design), which focus on reducing the error in the parameters, cannot be used. Replications would also be wasteful in cases when the computer simulation has no error. Criteria that are used to determine a good experimental design include integrated mean squared prediction error and distance based criteria .
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ICA on four randomly mixed videos Independent component analysis attempts to decompose a multivariate signal into independent non- Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results.
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All codes will have a probability of error greater than a certain positive minimal level, and this level increases as the rate increases. So, information cannot be guaranteed to be transmitted reliably across a channel at rates beyond the channel capacity. The theorem does not address the rare situation in which rate and capacity are equal. The channel capacity C can be calculated from the physical properties of a channel; for a band-limited channel with Gaussian noise, using the Shannon–Hartley theorem.
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To describe the internal motions of the spring connecting the two atoms, there is only one degree of freedom. Qualitatively, this corresponds to the compression and expansion of the spring in a direction given by the locations of the two atoms. In other words, ANM is an extension of the Gaussian Network Model to three coordinates per atom, thus accounting for directionality. The network includes all interactions within a cutoff distance, which is the only predetermined parameter in the model.
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Larger photocounts part has large noise dominated by count-dependent Gaussian noise and the small photocounts area dominated by the Poisson noise. For the accumulative emission smFRET data, the time trajectories contain mainly the following information: (1) state transitions, (2) noise, (3) camera blurring (analog of motion blur), (4) photoblinking and photobleaching of the dyes. The state transition information is the information a typical measurement wants. However, the rest signals interfere with the data analysis thus have to be addressed.
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Inscribed angles of a parabola A parabola with equation y = ax^2 + bx + c,\ a e 0 is uniquely determined by three points (x_1, y_1), (x_2, y_2), (x_3, y_3) with different x coordinates. The usual procedure to determine the coefficients a, b, c is to insert the point coordinates into the equation. The result is a linear system of three equations, which can be solved by Gaussian elimination or Cramer's rule, for example. An alternative way uses the inscribed angle theorem for parabolas.
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The ideas first appeared in physics (statistical mechanics) in the work of Pierre Curie and Pierre Weiss to describe phase transitions. MFT has been used in the Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–Fleer theory. Systems with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain Gaussian random-field theories, the 1D Ising model).
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Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.
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It also made advanced contributions to "fangcheng" or what is now known as linear algebra. Chapter seven solves system of linear equations with two unknowns using the false position method, similar to The Book of Computations. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns. The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution.
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The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is: : (sphere), where is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.
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The mixture model- based clustering is also predominantly used in identifying the state of the machine in predictive maintenance. Density plots are used to analyze the density of high dimensional features. If multi-model densities are observed, then it is assumed that a finite set of densities are formed by a finite set of normal mixtures. A multivariate Gaussian mixture model is used to cluster the feature data into k number of groups where k represents each state of the machine.
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Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck. The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.
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Riccati, Jacopo (1724) "Animadversiones in aequationes differentiales secundi gradus" (Observations regarding differential equations of the second order), Actorum Eruditorum, quae Lipsiae publicantur, Supplementa, 8 : 66-73. Translation of the original Latin into English by Ian Bruce. More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation.
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The use of Bayesian hierarchical modeling in conjunction with Markov chain Monte Carlo (MCMC) methods have recently shown to be effective in modeling complex relationships using Poisson-Gamma-CAR, Poisson- lognormal-SAR, or Overdispersed logit models. Statistical packages for implementing such Bayesian models using MCMC include WinBugs and CrimeStat. Spatial stochastic processes, such as Gaussian processes are also increasingly being deployed in spatial regression analysis. Model-based versions of GWR, known as spatially varying coefficient models have been applied to conduct Bayesian inference.
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As in quantum field theory the "fat tails" can 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. Fat tails can, however, also be due to other phenomena, such as a random number of terms in the central-limit theorem, or any number of other, non-econophysics models. Due to the difficulty in testing such models, they have received less attention in traditional economic analysis.
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Negentropy is defined as :J(p_x) = S(\varphi_x) - S(p_x)\, where S(\varphi_x) is the differential entropy of the Gaussian density with the same mean and variance as p_x and S(p_x) is the differential entropy of p_x: :S(p_x) = - \int p_x(u) \log p_x(u) \, du Negentropy is used in statistics and signal processing. It is related to network entropy, which is used in independent component analysis.P. Comon, Independent Component Analysis – a new concept?, Signal Processing, 36 287–314, 1994.
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The quotients are generally found by rounding the real and complex parts of the exact ratio (such as the complex number ) to the nearest integers. The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function is defined, which converts every Gaussian integer into an ordinary integer. After each step of the Euclidean algorithm, the norm of the remainder is smaller than the norm of the preceding remainder, .
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The cases and yield the Gaussian integers and Eisenstein integers, respectively. If is allowed to be any Euclidean function, then the list of possible values of for which the domain is Euclidean is not yet known. The first example of a Euclidean domain that was not norm-Euclidean (with ) was published in 1994. In 1973, Weinberger proved that a quadratic integer ring with is Euclidean if, and only if, it is a principal ideal domain, provided that the generalized Riemann hypothesis holds.
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The James–Stein estimator is a nonlinear estimator of the mean of Gaussian random vectors which can be shown to dominate, or outperform, the ordinary least squares technique with respect to a mean-square error loss function. Thus least squares estimation is not an admissible estimation procedure in this context. Some others of the standard estimates associated with the normal distribution are also inadmissible: for example, the sample estimate of the variance when the population mean and variance are unknown.
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Extra 'empty' dimensions are added to the source (known as the 'template' in this form of modelling), for example locating the 1D sound wave in 2D space. Further nonlinear dimensions are then added, produced by combining the original dimensions. The enlarged latent space is then projected back into the 1D data space. The probability of a given projection is, as before, given by the product of the likelihood of the data under the Gaussian noise model with the prior on the deformation parameter.
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1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, , but if units are included, is also equal to, say, among infinitely many similar "factorizations". 1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1).
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Developing and applying computational chemistry methods Curtiss helped develop the Gaussian-n series of quantum chemical methods for accurate energy calculations (G1, G2, G3, and G4 theories). These methods are for calculating the thermochemical properties of molecules and ions. Modeling lithium-ion batteries and beyond-lithium-ion batteries Curtiss is also involved in developing so-called "beyond-lithium- ion" batteries, such as lithium-sulfur and lithium–air batteries. He helped create a Li-O2 battery that runs on lithium superoxide.
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For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times. While exact models often scale poorly as the amount of data increases, multiple approximation methods have been developed which often retain good accuracy while drastically reducing computation time.
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The CALINE3 model is a steady-state Gaussian dispersion model designed to determine air pollution concentrations at receptor locations downwind of highways located in relatively uncomplicated terrain. CALINE3 is incorporated into the more elaborate CAL3QHC and CAL3QHCR models. CALINE3 is in widespread use due to its user friendly nature and promotion in governmental circles, but it falls short of analyzing the complexity of cases addressed by the original Hogan-Venti model. CAL3QHC and CAL3QHCR models are available in the Fortran programming language.
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Click to see animation. The evolution of an initially very localized gaussian wave function of a free particle in two-dimensional space, with color and intensity indicating phase and amplitude. The spreading of the wave function in all directions shows that the initial momentum has a spread of values, unmodified in time; while the spread in position increases in time: as a result, the uncertainty Δx Δp increases in time. The superposition of several plane waves to form a wave packet.
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The Independent Atom Model (abbreviated to IAM), upon which the Multipole Model is based, is a method of charge density modelling. It relies on an assumption that electron distribution around the atom is isotropic, and that therefore charge density is dependent only on the distance from a nucleus. The choice of the radial function used to describe this electron density is arbitrary, granted that its value at the origin is finite. In practice either Gaussian- or Slater-type 1s-orbital functions are used.
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This is very important for modeling chemical bonding, because the bonds are often polarized. Similarly, d-type functions can be added to a basis set with valence p orbitals, and f-functions to a basis set with d-type orbitals, and so on. Another common addition to basis sets is the addition of diffuse functions. These are extended Gaussian basis functions with a small exponent, which give flexibility to the "tail" portion of the atomic orbitals, far away from the nucleus.
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There the optical and mechanical modes hybridize and normal-mode splitting occurs. This regime must be distinguished from the (experimentally much more challenging) single-photon strong-coupling regime, where the bare optomechanical coupling becomes of the order of the cavity linewidth, g_0\geq\kappa. Effects of the full non-linear interaction described by \hbar g_0 a^\dagger a (b+b^\dagger) only become observable in this regime. For example, it is a precondition to create non- Gaussian states with the optomechanical system.
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This family of distributions is a special or limiting case of the normal- exponential-gamma distribution. This can also be seen as a three-parameter generalization of a normal distribution to add skew; another distribution like that is the skew normal distribution, which has thinner tails. The distribution is a compound probability distribution in which the mean of a normal distribution varies randomly as a shifted exponential distribution. A Gaussian minus exponential distribution has been suggested for modelling option prices.
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The Möbius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends. Finding algebraic equations cutting out a Möbius strip is straightforward, but these equations do not describe the same geometric shape as the twisted paper model above. Such paper models are developable surfaces having zero Gaussian curvature, and can be described by differential-algebraic equations. The Euler characteristic of the Möbius strip is zero.
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The Complete Basis Set (CBS) methods are a family of composite methods, the members of which are: CBS-4M, CBS-QB3, and CBS-APNO, in increasing order of accuracy. These methods offer errors of 2.5, 1.1, and 0.7 kcal/mol when tested against the G2 test set. The CBS methods were developed by George Petersson and coworkers, and they make extrapolate several single-point energies to the "exact" energy. In comparison, the Gaussian-n methods perform their approximation using additive corrections.
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In SURF, the lowest level of the scale space is obtained from the output of the 9×9 filters. Hence, unlike previous methods, scale spaces in SURF are implemented by applying box filters of different sizes. Accordingly, the scale space is analyzed by up-scaling the filter size rather than iteratively reducing the image size. The output of the above 9×9 filter is considered as the initial scale layer at scale s =1.2 (corresponding to Gaussian derivatives with σ = 1.2).
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Laser beams typically do not have sharp edges like the cone of light that passes through the aperture of a lens does. Instead, the irradiance falls off gradually away from the center of the beam. It is very common for the beam to have a Gaussian profile. Laser physicists typically choose to make the divergence of the beam: the far-field angle between the beam axis and the distance from the axis at which the irradiance drops to times the on-axis irradiance.
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In telecommunications, maximum-ratio combining (MRC) is a method of diversity combining in which: #the signals from each channel are added together, #the gain of each channel is made proportional to the rms signal level and inversely proportional to the mean square noise level in that channel. #different proportionality constants are used for each channel. It is also known as ratio-squared combining and predetection combining. Maximum-ratio combining is the optimum combiner for independent additive white Gaussian noise channels.
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We can utilize the output of numerical weather prediction models based on physical equations describing relationships in the weather system. Their predictive power tends to be less than, or similar to, purely statistical models beyond time horizons of 10–15 days. Ensemble forecasts are especially appropriate for weather derivative pricing within the contract period of a monthly temperature derivative. However, individual members of the ensemble need to be 'dressed' (for example, with Gaussian kernels estimated from historical performance) before a reasonable probabilistic forecast can be obtained.
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Image filtering (band pass filtering) is often used to reduce the influence of high and/or low spatial frequency information in the images, which can affect the results of the alignment and classification procedures. This is particularly useful in negative stain images. The algorithms make use of fast Fourier transforms (FFT), often employing gaussian shaped soft-edged masks in reciprocal space to suppress certain frequency ranges. High-pass filters remove low spatial frequencies (such as ramp or gradient effects), leaving the higher frequencies intact.
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He showed his formula to the mathematician Atle Selberg, who said that it looked like something in mathematical physics and that Montgomery should show it to Dyson, which he did. Dyson recognized the formula as the pair correlation function of the Gaussian unitary ensemble, which physicists have studied extensively. This suggested that there might be an unexpected connection between the distribution of primes (2, 3, 5, 7, 11, ...) and the energy levels in the nuclei of heavy elements such as uranium.John Derbyshire, Prime Obsession, 2004, .
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The series of positions as the asteroid moves across the sky allows the student to fit an approximate orbit to the asteroid. The measured asteroid coordinates (not the calculated orbital elements) are submitted to the Harvard-Smithsonian Center for Astrophysics. Over the decades SSP students have done their orbit determination calculations on mechanical calculators (1960s), then electronic calculators (1970s), then "mini-computers" (1980s), then personal computers (1990s and 2000s). In recent years they write their orbit determination programs in the Python programming language, employing the Gaussian method.
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In oscillators, however, the low-frequency noise can be mixed up to frequencies close to the carrier, which results in oscillator phase noise. Flicker noise is often characterized by the corner frequency fc between the region dominated by the low-frequency flicker noise and the higher-frequency "flat-band" noise. MOSFETs have a higher fc (can be in the GHz range) than JFETs or bipolar transistors, which is usually below 2 kHz for the latter. It typically has a Gaussian distribution and is time-reversible.
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In this type of network, each element in the input vector is extended with each pairwise combination of multiplied inputs (second order). This can be extended to an n-order network. It should be kept in mind, however, that the best classifier is not necessarily that which classifies all the training data perfectly. Indeed, if we had the prior constraint that the data come from equi-variant Gaussian distributions, the linear separation in the input space is optimal, and the nonlinear solution is overfitted.
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Since these distributions (which turn out to be approximately Gaussian) are directly related to the number of states, we may associate them with the entropy of the kink at any end-to-end distance. By numerically differentiating the probability distribution, the change in entropy, and hence free energy, with respect to the kink end-to-end distance can be found. The force model for this regime is found to be linear and proportional to the temperature divided by the chain tortuosity. Fig. 2 Isoprene backbone unit.
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The same optical effect can be achieved combining depth-of-field bracketing with multi exposure, as implemented in the Minolta Maxxum 7's STF function. In 2014, Fujifilm announced a lens utilizing a similar apodization filter in the Fujinon XF 56mm F1.2 R APD lens. In 2017, Sony introduced the E-mount full- frame lens Sony FE 100mm F2.8 STF GM OSS (SEL-100F28GM) based on the same optical Smooth Trans Focus principle. Simulation of a Gaussian laser beam input profile is also an example of apodization.
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Therefore, the time complexity, generally called bit complexity in this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a integer matrix is O(n^3) for the usual algorithms (Gaussian elimination). The bit complexity of the same algorithms is exponential in , because the size of the coefficients may grow exponentially during the computation. On the other hand, if these algorithms are coupled with multi-modular arithmetic, the bit complexity may be reduced to .
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The fundamental laser linewidth of light emitted from the lasing resonator can be orders of magnitude narrower than the linewidth of light emitted from the passive resonator. Some lasers use a separate injection seeder to start the process off with a beam that is already highly coherent. This can produce beams with a narrower spectrum than would otherwise be possible. Many lasers produce a beam that can be approximated as a Gaussian beam; such beams have the minimum divergence possible for a given beam diameter.
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CTF of the OAM microscope Choosing the optimum defocus is crucial to fully exploit the capabilities of an electron microscope in HRTEM mode. However, there is no simple answer as to which one is the best. In Gaussian focus one sets the defocus to zero, the sample is in focus. As a consequence contrast in the image plane gets its image components from the minimal area of the sample, the contrast is localized (no blurring and information overlap from other parts of the sample).
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The Klee–Minty cube has been used to analyze the performance of many algorithms, both in the worst case and on average. The time complexity of an algorithm counts the number of arithmetic operations sufficient for the algorithm to solve the problem. For example, Gaussian elimination requires on the order of D3 operations, and so it is said to have polynomial time-complexity, because its complexity is bounded by a cubic polynomial. There are examples of algorithms that do not have polynomial-time complexity.
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These histograms are computed from magnitude and orientation values of samples in a 16×16 region around the keypoint such that each histogram contains samples from a 4×4 subregion of the original neighborhood region. The image gradient magnitudes and orientations are sampled around the keypoint location, using the scale of the keypoint to select the level of Gaussian blur for the image. In order to achieve orientation invariance, the coordinates of the descriptor and the gradient orientations are rotated relative to the keypoint orientation.
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LINPACK 100 is very similar to the original benchmark published in 1979 along with the LINPACK users' manual. The solution is obtained by Gaussian elimination with partial pivoting, with 2/3n³ + 2n² floating point operations where n is 100, the order of the dense matrix A that defines the problem. Its small size and the lack of software flexibility doesn't allow most modern computers to reach their performance limits. However, it can still be useful to predict performances in numerically intensive user written code using compiler optimization.
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For random matrices with Gaussian distribution of entries (the Ginibre ensembles), the circular law was established in the 1960s by Jean Ginibre. In the 1980s, Vyacheslav Girko introduced an approach which allowed to establish the circular law for more general distributions. Further progress was made by Zhidong Bai, who established the circular law under certain smoothness assumptions on the distribution. The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Guangming Pan and Wang Zhou, and Friedrich Götze and Alexander Tikhomirov.
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Motivated by the central limit theorem, jitter can be modeled as a gaussian random variable. This suggests continually estimating the mean delay and its standard deviation and setting the playout delay so that only packets delayed more than several standard deviations above the mean will arrive too late to be useful. In practice, the variance in latency of many Internet paths is dominated by a small number (often one) of relatively slow and congested bottleneck links. Most Internet backbone links are now so fast (e.g.
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129 is the sum of the first ten prime numbers. It is the smallest number that can be expressed as a sum of three squares in four different ways: 11^2+2^2+2^2, 10^2+5^2+2^2, 8^2+8^2+1^2, and 8^2+7^2+4^2. 129 is the product of only two primes, 3 and 43, making 129 a semiprime. Since 3 and 43 are both Gaussian primes, this means that 129 is a Blum integer.
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MPQC (Massively Parallel Quantum Chemistry) is an ab initio computational chemistry software program. Three features distinguish it from other quantum chemistry programs such as Gaussian and GAMESS: it is open-source, has an object-oriented design, and is created from the beginning as a parallel processing program. It is available in Ubuntu and Debian. MPQC provides implementations for a number of important methods for calculating electronic structure, including Hartree-Fock, Møller-Plesset perturbation theory (including its explicitly correlated linear R12 versions), and density functional theory.
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One difference between Gaussian and SI units is in the factors of 4π in various formulas. SI electromagnetic units are called "rationalized",Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, " The Physics Teacher 24(2): 97–99. Alternate web link (subscription required) because Maxwell's equations have no explicit factors of 4π in the formulae. On the other hand, the inverse-square force laws - Coulomb's law and the Biot–Savart law - do have a factor of 4π attached to the r.
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The class of normal-inverse Gaussian distributions is closed under convolution in the following sense:Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013 if X_1 and X_2 are independent random variables that are NIG- distributed with the same values of the parameters \alpha and \beta, but possibly different values of the location and scale parameters, \mu_1, \delta_1 and \mu_2, \delta_2, respectively, then X_1 + X_2 is NIG-distributed with parameters \alpha, \beta, \mu_1+\mu_2 and \delta_1 + \delta_2.
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A new method was developed by F. Grimsley (AFWAL/FIBEC) to determine stress intensity, which used a 2-D Gaussian integration scheme with Richardson Extrapolation which was optimized by G. Sendeckyj (AFWAL/FIBEC). The resulting program was named MODGRO since it was a modified version of ASDGRO. Many modifications were made during the late 1980s and early 1990s. The primary modification was changing the coding language from BASIC to Turbo Pascal and C. Numerous small changes/repairs were made based on errors that were discovered.
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This is called a nested quadrature rule, and here Clenshaw–Curtis has the advantage that the rule for order N uses a subset of the points from order 2N. In contrast, Gaussian quadrature rules are not naturally nested, and so one must employ Gauss–Kronrod quadrature formulas or similar methods. Nested rules are also important for sparse grids in multidimensional quadrature, and Clenshaw–Curtis quadrature is a popular method in this context.Erich Novak and Klaus Ritter, "High dimensional integration of smooth functions over cubes," Numerische Mathematik vol.
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While the M2 factor does not give detail on the spatial characteristics of the beam, it does indicate how close it is to being a fundamental-mode Gaussian beam. It also determines the smallest spot size for the beam, as well as the beam divergence. M2 can also give an indication of beam distortions due to, for example, power-induced thermal lensing in the laser gain medium, since it will increase. There are some limitations to the M2 parameter as a simple quality metric.
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In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi- measure, or CSM) is a kind of prototype for a measure on an infinite- dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
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Gross's earliest mathematical worksIntegration and Nonlinear Transformations on Hilbert Space,Measurable Functions on Hilbert Space were on integration and harmonic analysis on infinite-dimensional spaces. These ideas, and especially the need for a structure within which potential theory in infinite dimensions could be studied, culminated in Gross's construction of abstract Wiener spaces in 1965. This structure has since become a standard framework Gaussian Measures in Banach Spaces, by Hui-Hsiung Kuo, An Introduction to Analysis in Wiener Space, by Ali S. Üstunel for infinite- dimensional analysis.
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A reference beam is a laser beam used to read and write holograms. It is one of two laser beams used to create a hologram. In order to read a hologram out, some aspects of the reference beam (namely its angle of incidence, beam profile and wavelength) must be reproduced exactly as when it was used to write the hologram. As a result, usually reference beams are Gaussian beams or spherical wave beams (beams that radiate from a single point) which are fairly easy to reproduce.
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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem. The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.
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There are two issues common to any approach at document layout analysis: noise and skew. Noise refers to image noise, such as salt and pepper noise or Gaussian noise. Skew refers to the fact that a document image may be rotated in a way so that the text lines are not perfectly horizontal. It is a common assumption in both document layout analysis algorithms and optical character recognition algorithms that the characters in the document image are oriented so that text lines are horizontal.
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In his original paper, Gauss made another choice, by choosing the unique associate such that the remainder of its division by is one. In fact, as , the norm of the remainder is not greater than 4. As this norm is odd, and 3 is not the norm of a Gaussian integer, the norm of the remainder is one, that is, the remainder is a unit. Multiplying by the inverse of this unit, one finds an associate that has one as a remainder, when divided by .
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