The similarity theory allows deriving non-trivial power-law scalings for the energy of fast electrons in underdense and overdense plasmas.
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The extraction of features are sometimes made over several scalings. One of these methods is Scale-invariant feature transform (SIFT) is a feature detection algorithm in computer vision; in this algorithm, various scales of an image are analyzed to extract features.
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Most physical systems possess natural symmetries (or invariance), i.e. there exist transformations (e.g. rotations, translations, scalings) that leave the system unchanged. From mathematical and engineering viewpoint, it makes sense that a filter well-designed for the considered system should preserve the same invariance properties.
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Most physical systems possess natural symmetries (or invariance), i.e. there exist transformations (e.g. rotations, translations, scalings) that leave the system unchanged. From mathematical and engineering viewpoints, it makes sense that a filter well-designed for the system being considered should preserve the same invariance properties.
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A further generalization is given by the Veronese variety, when there is more than one input variable. In the theory of quadratic forms, the parabola is the graph of the quadratic form (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form . Generalizations to more variables yield further such objects. The curves for other values of are traditionally referred to as the higher parabolas and were originally treated implicitly, in the form for and both positive integers, in which form they are seen to be algebraic curves.
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The multiplication of complex numbers can be visualized geometrically by rotations and scalings. The real numbers , with the usual operations of addition and multiplication, also form a field. The complex numbers consist of expressions : with real, where is the imaginary unit, i.e., a (non-real) number satisfying .
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In the "recipe" for image classification, groups of transformations are approximated with finite number of transformations. Such approximation is possible only when the group is compact. Such groups as all translations and all scalings of the image are not compact, as they allow arbitrarily big transformations. However, they are locally compact.
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The key idea is to smooth \chi_V a bit, by taking the convolution of \chi_V with a mollifier. The latter is just a bump function with a very small support and whose integral is 1. Such a mollifier can be obtained, for example, by taking the bump function \Phi from the previous section and performing appropriate scalings.
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0x7999999A The number of bits to store the fraction is 28 bits. Multiplying these 32 bit numbers give the 64 bit result This result is in B7 in a 64 bit word. Shifting it down by 32 bits gives the result in B7 in 32 bits. 0x1547AE14 To convert back to floating point, divide this by Various scalings may be used.
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The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Élie Cartan. Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are translations, rotations and scalings.
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Except for the initialization, all these changes of variable consists of the composition of at most two very simple changes of variable which are the scalings by two , the translation , and the inversion , the latter consisting simply of reverting the order of the coefficients of the polynomial. As most of the computing time is devoted to changes of variable, the method consisting of mapping every interval to is fundamental for insuring a good efficiency.
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Finally he left open the possibility of mixed phases, in which both are confined. Although such mixed phases had not been seen in quantum field theory at the time, they are now know to occur for example in the Argyres-Douglas conformal field theory. Therefore, he argued that gauge theories are necessarily in one of these four possible phases. 't Hooft found a simple formula for the scalings of the Wilson and 't Hooft operators in the various phases.
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There are many ways of representing the shape of an object. The shape of an object can be considered as a member of an equivalence class formed by taking the set of all sets of k points in n dimensions, that is Rkn and factoring out the set of all translations, rotations and scalings. A particular representation of shape is found by choosing a particular representation of the equivalence class. This will give a manifold of dimension kn-4.
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The Dutch organist and harpsichordist Class Douwes (circa 1650 – circa 1725) mentions instruments from nominal down to .Klaas Douwes, Grundig Ondersoek van de Toonen der Musijk (Franeker, 1699) The pitch differences between the models offered by the Ruckers workshops were by no means arbitrary, but corresponded to the musical intervals of a tone, a fourth, a fifth, an octave, and a ninth. Pitch assignments have been suggested for these instruments based on scalings provided by Douwes.Edwin M. Ripin, The "three foot" Flemish harpsichord.
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Binary scaling techniques were used in the 1970s and 1980s for real-time computing that was mathematically intensive, such as flight simulation and in Nuclear Power Plant control algorithms since the late 1960s. The code was often commented with the binary scalings of the intermediate results of equations. Binary scaling is still used in many DSP applications and custom made microprocessors are usually based on binary scaling techniques. The Binary angular measurement is used in the STM32G4 series built in CORDIC co-processors.
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The theory of operads is motivated by the study of loop spaces. A loop space ΩX has a multiplication :\Omega X \times \Omega X \to \Omega X by composition of loops. Here the two loops are sped up by a factor of 2 and the first takes the interval [0,1/2] and the second [1/2,1]. This product is not associative since the scalings are not compatible, but it is associative up to homotopy and the homotopies are coherent up to higher homotopies and so on.
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The Randomized Dependence CoefficientLopez-Paz D. and Hennig P. and Schölkopf B. (2013). "The Randomized Dependence Coefficient", "Conference on Neural Information Processing Systems" Reprint is a computationally efficient, copula-based measure of dependence between multivariate random variables. RDC is invariant with respect to non- linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence. For two binary variables, the odds ratio measures their dependence, and takes range non-negative numbers, possibly infinity: .
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That is, from the property :F(\lambda x_1,\dots, \lambda x_n)=\lambda^r F(x_1,\dots,x_n)\, it is possible to differentiate with respect to λ and then set λ equal to 1. This then becomes a necessary condition on a smooth function F to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the mathematical analysis considerations here). This setting is typical, in that there is a one- parameter group of scalings operating; and the information is coded in an infinitesimal transformation that is a first-order differential operator.
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For example, if we take U to be a line segment, then a continuous function is nothing but a curve in the complex plane, and we see that the zeta function encodes every possible curve (i.e., any figure that can be drawn without lifting the pencil) to arbitrary precision on the considered strip. The theorem as stated applies only to regions U that are contained in the strip. However, if we allow translations and scalings, we can also find encoded in the zeta functions approximate versions of all non-vanishing holomorphic functions defined on other regions.
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A tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed. The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference.
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In standard cosmology, matter is anything whose energy density scales with the inverse cube of the scale factor, i.e., This is in contrast to radiation, which scales as the inverse fourth power of the scale factor and a cosmological constant, which is independent of a. These scalings can be understood intuitively: For an ordinary particle in a cubical box, doubling the length of the sides of the box decreases the density (and hence energy density) by a factor of 8 (= 2). For radiation, the energy density decreases by a factor of 16 (= 2), because any act whose effect increases the scale factor must also cause a proportional redshift.
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In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape. Mathematician and statistician David George Kendall writes: > In this paper ‘shape’ is used in the vulgar sense, and means what one would > normally expect it to mean. [...] We here define ‘shape’ informally as ‘all > the geometrical information that remains when location, scaleHere, scale > means only uniform scaling, as non-uniform scaling would change the shape of > the object (e.g.
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Over the following two decades, Goldston led several experimental efforts studying the physics and efficacy of heating tokamak plasmas with neutral beams, discovering along the way a type of instability that could eject energetic beam ions if the neutral beam system was aimed too orthogonally with respect to the tokamak plasma. He also explored a number of other loss mechanisms for energetic ions. This proved crucial in determining the range of angles over which future neutral beam systems could access toroidal plasma configurations. Drawing upon a wide body of experimental data from most of the tokamaks then operating, Goldston developed the first widely applicable empirical scaling relationship for the confinement of energy in tokamak plasmas as a function of such parameters as the major radius, minor radius, density, current, and heating power from such sources as neutral beam systems. This scaling relationship, which came to be known as “Goldston Scaling,” provided a predictive tool for estimating the performance of tokamaks, and found wide utility, eventually forming the starting point for later energy confinement scalings based upon much larger analyses of data from successive generations of tokamaks.
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