Ax and Bx … 4(5) can. C Bivariate and Multivariate Normal Integrals 417 C.1.2 ReductionsofForm Although(C.5)canbeintegratednumerically,italsocanbereducedtoasingleintegral Then we keep going for degrees up to three. So first we substitute x to power zero, which is just one. / integral 1 is done by squaring the integral, combining. Basic Integration Formulas List Of Integral Formulas from cdn1.byjus.com The difficulties with primitive gto's that contribute to more than one ao increase down the periodic table. Elliptic functions and integrals ... weights, and total weights for the appropriate form of Gaussian quadrature. Gaussian quadrature chooses the points for evaluation in an optimal, rather than equally-spaced, way. This Demonstration shows how substitution, integration by parts, and algebraic manipulation can be used to derive a variety of reduction formulas. with. Gaussian Integrals. Also, integrals of the Q-function are evaluated. !Φ(x)+C{\displaystyle \int x^{2k+2}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{2j+1}+(2k+1)! Gaussian quadrature¶. variables. n. even. These restrictions are shown in the third column. The definite integral, which gives the area under a curve between two points. T1 - Efficient simulations for the exponential integrals of Hölder continuous Gaussian random fields. Gaussian quadratures are developed for the evaluation of the integrals given in the title. the. The theorem THE GAUSSIAN INTEGRAL KEITH CONRAD Let I= Z 1 1 e 21 2 x dx; J= Z 1 0 e 2x dx; and K= Z 1 1 e ˇx2 dx: These numbers are positive, and J= I=(2 p 2) and K= I= p 2ˇ. Numerical differentiation of forces/dipoles to obtain polarizability/ hyperpolarizability. 13 Functional Derivative 1. done. Some of the integrals are generalizations of earlier results, but others are new; all de- Regarded as a function of the complex variable $ z $, the probability integral $ \mathop{\rm erf} ( z) $ is an entire function of $ z $. After an affine change of variables of the form ˜x = Ax+c, this amounts to performing an integral of the form I[f]= Rn (1.1) e−xT xf(x)dx. !\,\Phi (x)+C} The overlap of two Gaussian states. exponent. collapse all. Processes route section, builds list of links to execute, and initializes scratch files. Other integrals can be approximated by versions of the Gaussian integral. Iwasawa, gaussian integral puzzle, math. / integral 1 is done by squaring the integral, combining. Table of Integrals BASIC FORMS (1)!xndx= 1 n+1 xn+1 (2) 1 x!dx=lnx (3)!udv=uv"!vdu (4) "u(x)v!(x)dx=u(x)v(x)#"v(x)u! 2. We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Linear subspace property of Gaussian integrals. have Gaussian distributions. Integrals with Trigonometric Functions (71) Z sinaxdx= 1 a cosax (72) Z sin2 axdx= x 2 sin2ax 4a (73) Z sin3 axdx= 3cosax 4a + cos3ax 12a (74) Z sinn axdx= 1 a cosax 2F 1 1 2; 1 n 2; 3 2;cos2 ax (75) Z cosaxdx= 1 a sinax (76) Z cos2 axdx= x 2 + sin2ax 4a (77) Z cos3 axdx= 3sinax 4a + sin3ax 12a 8 be. The indefinite integral, which is a family of functions formed by adding an arbitrary constant to an antiderivative. A few functions are also provided in order to perform simple Gaussian quadrature over a fixed interval. be. In this talk at 19:32, the following method of reducing a multivariate integral is discussed: Let θ ∈ R p be the parameter vector, x i ∈ R p be the i -th data vector, y i ∈ { − 1, 1 }, be the corresponding binary observation. e−ax2dx= 1 2 π a # $% & ’(1 2 0 ∞ ∫ ax xe−2dx= 1 2a 0 ∞ ∫ x2e−ax2dx= 1 4a π a # $% & ’(1 2 0 ∞ ∫ x3e−ax2dx= 1 2a2 0 ∞ ∫ x2ne−ax2dx= 1⋅3⋅5⋅⋅⋅(2n−1) 2n+1an π a $ %& ’ 1 2 0 ∞ ∫ x2n+1e−ax2dx= n! The book describes. $\begingroup$ @CarloBeenakker I first developed a probabilistic code working with multivariate Gaussian distributions with pd covariance matrices. PY - 2014/2. Gaussian processes can be used as priors on functions. square. With notation as above, I= p 2ˇ, or equivalently J= p ˇ=2, or equivalently K= 1. Keywords: Gaussian processes, Malliavin calculus, multiple stochastic integrals, Non-central limit theorems, weak convergence Rights: This work is licensed under … use. Molecular Integrals. the Lebesgue measure but absolutely continuous w.r.t the restriction of the Lebesgue measure on the … Then we substitute x to power one, the integral is zero, and again we have an expression for the quadrature. (a) Suppose f(z) is de ned in the upper half-plane. then. Gaussian 09 Links. In addition, we also show how one can arrive at the generalized star product using ordinary Gaussian (bosonic) distributions. 3 is half of Eq. Proof of Gaussian Intergral May. ∫ 0 ∞ e − α x 2 d x = 1 2 π α {\displaystyle \int _ {0}^ {\infty }e^ {-\alpha x^ {2}}\mathrm {d} x= {\frac {1} {2}} {\sqrt {\frac {\pi } {\alpha }}}} is a result that can be used to find numerous related integrals. These are dealing These fluctuations also modify the saddle point free energy. The Gaussian integral, also called the probability integral and closely related to the erf function, is the integral of the one-dimensional Gaussian function over (-infty,infty). So far, so good. A Brief Look at Gaussian Integrals WilliamO.Straub,PhD Pasadena,California January11,2009 Gaussianintegralsappearfrequentlyinmathematicsandphysics. If it's integrated, the result is two, and the quadrature rule is just the sum of the weights. Double numerical differentiation of energies to produce frequencies. by. from now on we will simply drop the range of integration for integrals from −∞ to ∞. a. wedge. over. Since xis an odd function, any Then I realized that I need to work with psd covariances matrices. Where is this proof of the Gaussian integral being zero wrong? Y1 - 2014/2. Examples. (As an aside, although you do not need to learn this derivation, this is how one can derive the basic Gaussian integral. (x)dx RATIONAL FUNCTIONS (5) 1 ax+b!dx= 1 a ln(ax+b) (6) 1 (x+a)2!dx= "1 x+a (7)! CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We present a novel formula for Gaussian Berezin correlation functions. Gaussian quadrature¶. and observed that Gaussian is the oddball in including the factors coming from permutational symmetry ("redundancy factors"). 1: All integrals are done as regular integrals. In this paper, we provide a new way of obtaining the generalized star product. Let’s consider the overlap integral of two 1D Gaussians, a a and b b. Sab = ∫ Gi(x,α,Ax)Gj(x,β,Bx)dx = ∫ KABxi Axj Bexp(−px2 P)dx S a b = ∫ G i ( x, α, A x) G j ( x, β, B x) d x = ∫ K A B x A i x B j exp ( − p x P 2) d x. where we used the Gaussian product theorem so that. Since the gaussian integral is a definite integral and must give a constant value a second definition, also frequently called the euler integral, and already presented in … In quantum eld theory, gaussian integrals come in two types. 1. Gaussian Integral (formula and proof) - SEMATH INFO from semath.info Jameson, the probability integral by volume of. Then perform the Gaussian integrals separately for each term in the sum, I = (N ∏ i = 1∫e − βAidJii)(N ∏ i < j = 1∫∫e − βBijdJijdJji) = (π / β)N2 / 2(N ∏ i = 1(λi − τ) − 1 / 2)(N ∏ i < j = 1(λiλj − τ2) − 1 / 2), where I have defined β = 1 2N(1 − τ2) − 1, and assumed that β > 0, λi > τ for all i. C Bivariate and Multivariate Normal Integrals 417 C.1.2 ReductionsofForm Although(C.5)canbeintegratednumerically,italsocanbereducedtoasingleintegral equation. import scipy.integrate as integrate import numpy as np def gaussian (x, mu, sig): norm = 1/np.sqrt (2*np.pi*sig*sig) return norm * np.exp (-np.power (x - mu, 2.) 9 Free Particle and Harmonic Oscillator . p = a + b. μ = a b a + b. P x = a A x + b B x p. X A B = A x − B x. 2an+1 0 ∞ ∫ xne−axdx= n! angle. First of all, notice that all \odd-ordered" Gaussian integrals are zero: Z 1 1 x2n+1e x2dx= 0 (4) 1. The well-known Gaussian integral can be evaluated in closed form, even though there is no elementary function equal to the indefinite integral . "A hybrid Gaussian and plane wave density functional scheme." integrating. / (2. 6 Gaussian Functional Integrals . Computing higher-order Gaussian integrals uses an equally clever trick. 7 From Path Integration to the Schrödinger Equation . Some integrals are presented that can be expressed in terms of the Q-function, which is defined as lý2 +a2\ Q(a, b) f dx x exp 2 1. taking. by. Here a and b are Gaussian exponent factors. basic integral we need is G ≡ Z ∞ −∞ dxe−x2 The trick to calculate this is to square this using integration variables x and y for the two integrals and then evaluate the double integral using polar coordinates. This results from the approximation of four-center two-electron integrals by corresponding three-center integrals. 10. A particular realization of Gaussian path integrals is given by "white noise functionals". the Gaussian integrals, but want to see a readable computer implementation. ( i m 2 ℏ ⋅ 2 ϵ ( x 2 − x 0) 2). Although more functions and more integrals now are part of the calculation, the integrals involving Gaussian functions are quicker to compute than those involving exponentials, so there is a net gain in the efficiency of the calculation. Indefinite integral . The asymptotic representation for large … These are returned in an n x 3 array with roots in the first column, weights in the second column, and total weights in the final column. N.B. So G2 = Z dxe−x2 Z dye−y 2= Z dx Z dye−(x2+y ) = In the paper we consider the density functions of random variables that can be written as integrals of exponential functions of Gaussian random fields. List of integrals of exponential functions 2 where where and is the Gamma Function when , , and when , , and Definite integrals for, which is the logarithmic mean (the Gaussian integral) (see Integral of a Gaussian function) (!! Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. So if we approximate f by some polynomial, and keep the weighting function as it is, we will have a quadrature where the weights, the coefficients, depend on essentially the moments of the weighting function the integrals of this form. 11 Path Integral Perturbation Theory . (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! All integrals done as Raffenetti if there are N or more matrices; all as regular if there are less than N. 0: Default: let FoFJK decide. Integral. release(expr) evaluates the integrals in the expression expr. However, apparently I've misunderstood something. Integral. Consider the integral below. Springer, 1998, p. 112. Light emission of a molecule formed from two atoms. Gaussian sampling with an integral gram root. Which is an integral seen often in statistics, because f (x) = … The results are then represented as areas under a curve (shown in red). 1. Gaussian integrals in Feynman and Hibbs. Berny optimizations to minima and TS, STQN transition state searches. N2 - In this article, we consider a Gaussian random field f (t) living on a compact set T ⊂ Rd and the computation of the tail probabilities P(∫ T … Third, for improper integrals, this book emphasizes the Parameter selections of the integral gram decomposition. Integral 2 is done by changing variables then using integral 1. II.Gaussian G ntegrals I In the previous section, the energy cost of fluctuations was calculated at quadratic order. Suppose we want I= Z +1 1 exp x2 dx: Then we square this: I2 = Z +1 1 exp x2 in. 2. The RI (resolution of the identity) technique achieves significant increases in efficiency for various molecular electronic structure methods. Troublesome integrals in Hamiltonian matrix elements for a system of two interacting electrons. Integration, in that case, is taken over the space corresponding to the second Gaussian. • using integration by parts on the fourier transform of f , with u = f (x) and dv = e−ixy dx, we obtain (f(f ))(y) = iy(ff )(y). Students rst learn how to evaluate this integral in the case where a is a real, positive constant. 10 Matrix Element of a Physical Operator via Functional Integral . In this chapter we will give an introduction to definite and indefinite integrals. Unevaluated Integral. General idea: Lippert, Gerald, et al. The need for a flexible, principled, probabilistic model of functional relations is common in practice. (Other lists of proofs are in [4] and [9].) We will give multiple proofs of this result. 1. Fourier integrals are also considered. The gaussian integral is dened to be: List of integrals of exponential functions. 2. [2] See List of statements undecidable in ZFC. It's DFT, but just look for the Coulomb integrals. The line integral, an integral under a curve, taken over a scalar or a vector field. The weights and abscissae for the semi-infinite integral are given for two These integrals turn up in subjects such as quantum field theory. The Gaussian integral. and. Integration is the process of finding a function with its derivative. The graph of the probability integral and its derivatives are illustrated in the figure. In the equations in this section the following definitions is used. Our formula allows one to evaluate these integrals without calculating the inverse of the matrix. as a Gaussian integral. The difficulties with primitive gto's that contribute to more than one ao increase down the periodic table. $\endgroup$ – nOp Jun 12 '20 at 22:31 $\begingroup$ @nOp, just project the higher dimensional Gaussian to the subspace (center and covariance matrix) and use the above. First Lesson: Gaussian Integrals Given the experience accumulated since Feynman’s doctoral thesis, the time has come to extract a simple and robust axiomatics for func-tional integration from the work done during the past sixty years, and to investigate approaches other than … In this Demonstration, we perform numerical integrations for , with and . A strategy for an efficient evaluation of molecular integrals involving gaussian functions of high angular momentum is reported. Before calculating this modification, we take a short (but necessary) mathematical diversion on performing Gaussian integrals. Integral. AU - Liu, Jingchen. The linear combination of primitive Gaussian functions is called a contracted Gaussian function. Integrals with Trigonometric Functions Z sinaxdx= 1 a cosax (63) Z sin2 axdx= x 2 sin2ax 4a (64) Z sinn axdx= 1 a cosax 2F 1 1 2; 1 n 2; 3 2;cos2 ax (65) Z sin3 axdx= 3cosax 4a + cos3ax 12a (66) Z cosaxdx= 1 a sinax (67) Z cos2 axdx= x 2 + sin2ax 4a (68) Z cosp axdx= 1 a(1 + p) cos1+p ax 2F 1 1 + p 2; 1 2; 3 + p 2;cos2 ax (69) Z cos3 axdx= 3sinax 4a + sin3ax 12a (70) Z cosaxsinbxdx= cos[(a b)x] 2(a b) cos[(a+ … A formula well known in the literature expresses these quantities in terms of submatrices of the inverse matrix appearing in the Gaussian action. Integral can refer to one of several closely related concepts from calculus. Lines such as those shown below would likely give much better approximations in most cases. done. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity. The Gaussian integration is a type of improper integral. List of integrals of exponential functions. 2. The rst theorem is for functions that decay faster than 1=z. Feynman diagrams in Gaussian integrals. 7. can. ∫x2k+2ϕ(x)dx=−ϕ(x)∑j=0k(2k+1)!!(2j+1)!!x2j+1+(2k+1)! The release function ignores the 'Hold' option in the int function when the integrals are defined. Theorem 9.1. changing. Also let ϕ ( ⋅) denote the logistic sigmoid function. Table of Useful Integrals, etc. The calculation of mean or variance of certain outputs under random inputs requires the evaluation of certain integrals. I think that involves a mapping between Gaussian and PW basis. 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. $\endgroup$ – Jarek Duda Jun 13 '20 at 5:06 Gaussian integrals in Feynman and Hibbs. Ask Question Asked today. Gaussian Quadratures for the Integrals / exp( — x2)f0x)dx and / exp( — x2)fix)dx Jo Jo By N. M. Steen, G. D. Byrne and E. M. Gelbard Abstract. See also. 8 Quasi-Classical Approximation . Understanding two-electron integrals in Gaussian 09. These begin with the two basic formulas, change of. Hot Network Questions 1. the integral by I, we can write I2 = µZ ∞ −∞ e−x2 dx ¶ 2 = Z ∞ −∞ e−x2 dx ∞ −∞ e−y2 dy (2) where the dummy variable y has been substituted for x in the last integral. The following is a list of integrals of exponential functions. It is shown that the three-center integrals require For. π. Viewed 5 times 0 $\begingroup$ I'm relatively new to complex analysis, but have been diving into it since it's been proving useful in tackling some of my recent problems. The product of two integrals can be expressed as a double integral: I2 = Z ∞ −∞ Z ∞ −∞ e−(x2+y2) dxdy The differential dxdy represents an elementof area in cartesian The n + p = 0 mod 2 requirement is because the integral from −∞ to 0 contributes a factor of (−1) n+p /2 to each term, while the integral from 0 to +∞ contributes a factor of 1/2 to each term. List of integrals of Gaussian functions Original algorithms and computational methods are discussed, which have been incorporated into the integral part of the molecular SCF program system ASTERIX. Both types of integrals are tied together by the fundamental theorem of calculus. 4(−π. The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to . SEMATH INFO. – The function f (x) has points of discontinuity in the interval [a,b]. (ax) 'b/ and where 10 is the modified Bessel function of order zero. This leaflet provides such a table. What you have is a MultinormalDistribution.The quadratic and linear forms in the exponential can be rewritten in terms of $-\frac12(\vec{x}-\vec{\mu})^\top\Sigma^{-1}(\vec{x}-\vec{\mu})$ where $\vec{\mu}$ represents the mean and $\Sigma$ the covariance matrix, see the documentation.. With this, you can do integrals of the type given in the question by invoking Expectation, as in this … List of integrals of exponential functions. * sig*sig)) def gaussianGreater (x, mu, sig): Integrand = lambda x: gaussian (x, mu, sig) return integrate.quad (Integrand,-np.Inf, x) [0] My problem now lies in the integration bounds of my gaussianGreater function while it is being … completing. 12 Transition Matrix Element . by. Gaussian integration is simply integration of the exponential of a quadratic. 3. is. Note: The improper integrals in formulas 2, 3 are convergent if the upper and lower limits exist and are finite. 2nd, 2019. !Φ(x)+C We will discuss the definition and properties of each type of integral as well as how to compute them including the Substitution Rule. done. I’ll talk a bit about some computational considerations at the end, but my goal is to convert equations to code. Integrals of polynomials We can see this by drawing the second function and because integrals are-4 -2 0 2 4 x-1-0.5 0 0.5 1 Figure 2: areas under a curve it becomes obvious that Eq. List of Gaussian Keywords # ADMP; BD; BOMD; CacheSize; CASSCF; CBS Methods; CBSExtrapolate; CCD and CCSD; Charge; ChkBasis Open Live Script. Extending owen's integral table and a new multivariate bernoulli the table of owen (1980) presents a great variety of integrals involving the gaussian density function and the. References [1] S. Srivastava A course on Borel sets. Molecular Integrals over Gaussian Basis Functions 181 4.3 Generation of the [03 (m) Integrals Given a batch of shell-quartets, the real computational work (denoted To in Figures 1 and 2) can begin. Sometimes restrictions need to be placed on the values of some of the variables. Multiple Wiener integrals and stochastic integrals are defined for Gaussian processes, extending the related notions for the Wiener process. More generally, integrals of the form can be evaluated for positive integers [1]. The first is fixed_quad, which performs fixed-order Gaussian quadrature.The second function is quadrature, which performs Gaussian quadrature of multiple orders until the difference in the integral estimate is beneath some tolerance supplied by the user. Gaussian integral One application of Fubini's theorem is the evaluation of the Gaussian integral which is the basis for much of ... [0, 1] does hold, and whenever the two iterated integrals exist they are equal. 1. I was comparing the two-electron integrals (electron repulsion integrals, ERI) printed by various well-established packages (Gaussian, GAMESS, Molpro etc.) Express j2 as a double integral and then pass to polar coordinates: A table of normal integrals. The first is fixed_quad, which performs fixed-order Gaussian quadrature.The second function is quadrature, which performs Gaussian quadrature of multiple orders until the difference in the integral estimate is beneath some tolerance supplied by the user. Integrals of Airy functions. Nonetheless their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral Gaussian function-Wikipedia for some real constants a, b, c > 0 can be calculated by putting it into the form of a Gaussian integral . Thus, an improper integral is an integral over an unbounded interval or of an unbounded function. Gaussian integrals and derivative integrals, and then show how a change of differentiation variables simplifies the formulas. This is because the Gaussian factor, e x2, is an even function. The Annals of Probability. We summarize formulas of the Gaussian integral with proofs. And the result of this integration is given as. Properties of Improper Integrals. ∫x2k+2ϕ(x)dx=−ϕ(x)∑j=0k(2k+1)!!(2j+1)!!x2j+1+(2k+1)! Consequently, such an approach is demonstrably useful in a large variety of applications. Theorem. 1. Basic integration formulas on different functions are mentioned here. Elementary integrals all of these follow immediately from the table of 2. "1 (8)!x(x+a)ndx= (x+a)1+n(nx+x"a) (n+2)(n+1) (9) dx!1+x2 =tan"1x (10) dx!a2+x2 = 1 a tan"1(x/a) (11) xdx!a2+x2 = 1 2 ln(a2+x2) (12) x2dx!a2+x2 … Contains the information about how the integrals are calculated. A selection of more complicated integrals. For example,, since the derivative of is . Two challenges of Gaussian process modelling are often encountered. Indefinite integrals are antiderivative functions. Gaussian Integration: Optimal integration points But this is not likely the best line for approximating the integral. 9.2 Integrals of functions that decay The theorems in this section will guide us in choosing the closed contour Cdescribed in the introduction. Active today. In the first stage of this, the seven basic shellquartet parameters 202 = 1 / [ & + & ] 2T = 2192 R 2 (83) are constructed. Part 2of 3:Generalizations Download Article. We will give the Fundamental Theorem of Calculus showing the relationship between derivatives and integrals. I'm not sure if anyone has done it, but the closest I can think of is the GPW (mixed Gaussian Plane Wave) method. A few functions are also provided in order to perform simple Gaussian quadrature over a fixed interval. Rationale . Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. to. the. AU - Xu, Gongjun. The method is based on the Grassmannian (fermionic) integration theory given by Berezin and it evaluates certain Gauss–Grassmann integrals. I am trying to compute an integral, as part of the expected value formula (using a Gaussian PDF) [tex]\int_{-∞}^{∞} (x)^2 p(x) dx [/tex] Where p(x) is the Gaussian probability density function: [tex]\frac{1}{\sigma \sqrt(2 \pi)} \exp(\frac{-x^2}{2 \sigma^2})[/tex] My aim after this is to be able to compute for all even x^n in the above formula. N: Integrals with degree of contraction greater than or equal to N are done at regular integrals. integrals or have been presented only for some special casesof parameters or for integration only along the real axis of the complex plane. The following is a list of integrals of exponential functions. In quantum eld theory, gaussian integrals come in two types. I was going through the calculation of the free-particle kernel in Feynman and Hibbs (pp 43). What you have is a MultinormalDistribution.The quadratic and linear forms in the exponential can be rewritten in terms of $-\frac12(\vec{x}-\vec{\mu})^\top\Sigma^{-1}(\vec{x}-\vec{\mu})$ where $\vec{\mu}$ represents the mean and $\Sigma$ the covariance matrix, see the documentation.. With this, you can do integrals of the type given in the question by invoking Expectation, as in this … Traditional numerical integration methods applied to the integral normally lead to poor accuracy due to the rapid change in high order derivatives of its integrand when the standard deviation is small. So, whether this whole idea just works or not depends on whether we can compute those integrals easily. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. In this case the distributions are NOT absolutely continuous w.r.t. Apart from the formulas for integration, classification of integral formulas and a few sample questions are also given here, which you can practice based on the integration formulas mentioned in this article. In fact, I’ve tried to structure the equations and the code in such a way that the two look very similar. The use of reduction formulas is one of the standard techniques of integration taught in a first-year calculus course. Second, many of the integrals presented here cannot be obtained using a computer (except via an approximate numerical integration). We will also discuss the Area Problem, an important interpretation …

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