| 词汇 | example_english_integral |
| 释义 | Examples of integralThese examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. Indeed, since the particle radius is small, the related contribution of the correlation function to all relevant integrals should be negligible. The system has the following integrals of motion. In this case we find exact solutions in terms of elliptic integrals of the fir st kind. There must be two linearly independent particular integrals, because the differential equation (46) is of order two. Results are expressed in terms of integrals involving eigenfunctions and trapped oscillation field structures, although explicit expressions for these cannot be given. Taking this approximation into account, we can easily calculate the integrals in (16). The equations of motion admit the six known integrals. The sound radiation associated with the instability wave can be found by evaluating the integrals of equation (3.20). The power integrals are obtained by respectively multiplying by ys and vi. One might expect that a definition based on certain integrals of the temperature profile would better describe the stability of such complex systems. 2 (because of the strong weighting factors eq and e2qin the integrals). However, such manipulation serves little purpose, since values for @areobtained easily enough by direct numerical evaluation of the integrals. The surface and volume integrals are computed in a manner analogous to those for the six-sided elements. The integrals which do not involve the basic state can be evaluated exactly. Evaluation of the integrals for the cylindrical and spherical cases would yield profiles similar to the ones shown in figures 4-9. The double integrals required in (2.8) cannot be performed analytically. The integrals on the right-hand side are identical, for w is a continuous function. I n such cases, the azimuthal (8) integration can be performed analytically and the surface integrals are reduced to line integrals. Using this to evaluate the branch-cut integrals one gets (38). Here it has been assumed that the disturbance is sufficiently localized such that the integrals (9), (10) exist. In the case of geodesically equivalent billiard tables, these integrals are pairwise commuting. The purpose of the present subsection is to develop the necessary change of variables between integrals in these two coordinate systems. We also approximate the boundary integrals of (3.11) by the trapezoidal rule. First we make some observations concerning integrals of the type (3.135). All circulation integrals around loops larger than about one grid square are conserved however. In fact, they are enfolded into the integrals corresponding to diagrams 2 and 10, respectively. Commuting integrals of the geodesic flow are just lifts of commuting functions in the space of geodesics. The results are then obtained by bounding the line u integrals (4.3) and (4.4), which represent an example of nontrivial technical work. Examples of such features are symplecticity, volume preservation, integrals, symmetries, and many more. They arise naturally in many stochastic optimization problems defined in terms of continuous random variables, since expectations or probabilities involving these are given by integrals. We therefore again seek weak solutions, where the behaviour of infinitesimally small fluid volumes is ignored, and work only with integrals. In this section we discuss another method of how to deal with oscillatory integrals. In contrast, it can have an arbitrarily large number of weak integrals. We show how to use an adjoint symmetry and functions of known first integrals to obtain new first integrals. We begin our consideration by calculating the integrals in (6), (7). We show that for the integrals under consideration these hyperasymptotic expansions can be made accurate to any desired exponentially-small order. A method of finding integrals for three-dimensional dynamical systems. In the case of fluid particles, further integrals over the volume and surface of the particle are required. The volume integrals may also be evaluated if it is supposed that the eddy is a cylinder centred on the edge of the half plane. The first two can be reduced to double definite integrals, while the last two involve both double and triple integrals. The other two integrals are still defined only in the finite part sense. One deals accordingly with integrals or sums over t-values. The trajectories of absolute motion of vortices remain axisymmetric during the entire evolution owing to constraints imposed by the integrals of motion. In the seven integrals in (4.1) the integration variables and are involved implicitly in the first argument of the steady source distribution function. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. Recursion relations between the integrals are derived in order that only the simplest finite integrals need to be evaluated directly. Under the assumption of non-resonant interaction of the particle with the radiofrequency field, these integrals represent adiabatic invariants of the particle motion. The final integrals in the fir st terms of (72) and (73), written out in (75) and (76), are also manifestly free of pinching singularities. The integrals in (30) were calculated in the same way. We present a novel application in computing integrals in dynamical systems. We can avoid a difficult calculation of some integrals using the following integration paths. In this case, one may extend (3.10) to z > 1 by writing the relevant integrals in terms of the exponential functions with a complex argument. Let 1 and 2 be integrals of 1 and 2 within the boundary layer, respectively. In other words, one can construct an example where the integrals (4.2) are all zero but condition three in the main theorem is not true. We turn now to the problem of transfer of orbital integrals. We need the following general results on the integrals of regular conditional probability distributions and their measurability later in this paper. In the limit we obtain the integrals over the interval t (0, 1). Nonoscillation of elliptic integrals related to cubic polynomials with symmetry of order three. Velocities and integrals of forward flow in systole, early diastole and late diastole were measured; as was duration of pulmonary regurgitation. The integrals in equation (4.5) may then be evaluated analytically. However, in the present analysis, it is the eigenfunction that must be calculated in order to evaluate the integrals required in the multiple-scales asymptotic solution. The equivalence means that the integrals of both sides with respect to every -invariant measure on coincide. In general, we have been able to produce a set of tame integrals. Numerical estimation of integrals was performed by a simple dot product between sampled spectra. The daily integrals of the directly measured light in the understorey (mol photons m-2 d-1) were averaged over the entire 9-mo period. We present an explicit construction formula to find the resulting first integrals in terms of integrating factors, and discuss techniques for finding integrating factors. Integration by parts removes the derivatives of the unknown from the integrals. By using the leading-order solution, we may explicity evaluate these integrals. Many of these equations could be put in a form where the integrals appear as the eigenvalues of a linear operator associated with the solution. We show that the geodesic flow of a metric all of whose geodesics are closed is completely integrable, with tame integrals of motion. If this new set can be constructed so that the integrals can be calculated or estimated, then actual convergence rates follow. Such definite integrals are best evaluated using integration in the complex plane. Exact integrals of motion have been used to determine an exact solution and new approximate solutions of the equation of motion. Note that some integrals are positive or zero, depending on the magnitude of one of their terms. To evaluate the velocity-space integrals, it is necessary to provide the actual distribution function, which is a function of the constants of motion only. The free functions are thus at our disposal, playing the role of integrals of the motion in the integration of a usual differential system. Since (4.9) contains integrals of the resistivity and entropy terms in the energ y equation, they both determine the leading-order isothermal solution. In this case the two integrals together cannot be carried out analytically. From this equation, one can indeed analytically calculate the integrals that define the solutions. We shall denote these integrals by (n), where 1 n(r, t) d#r. The vorticity evolution equation implies that the spatial integrals of all moments of the vorticity are conserved. In addition, it enables to calculate the integrals in the dialectric permeability tensor analytically. The manner in which the flux surfaces foliate space is determined by complete elliptic integrals with the modulus playing the role of the foliation parameter. In this approximation, we can give general expressions for the solution of linearized, high-frequency oscillations in such a device in terms of two integrals. A quadrature scheme generates an approximation of the integrals in, where we employ exclusively the values from. Here we introduce a modified trapezoid rule designed specifically for the singular integrals at hand. In fact, the two integrals in the solvability condition are independent of x0 and the shape of the domain. The contribution from some parts of this boundary can be explicitly evaluated, and the other contributions are evaluated as integrals in the -plane. Now we deal with the two integrals separately. The difficulty in settling these questions stems from the fact that one deals with complicated multidimensional oscillatory integrals. Unfortunately, attempts to show that integrals of this form are non-zero through analytic means alone have so far been unsuccessful. The elements of the resulting stiffness matrix are integrals, which have to be numerically evaluated over various regions. Conventional algorithms for the numerical computation of such integrals are usually limited by the curse of dimensionality. The degrees of freedom are integrals of an l-form weighted with polynomials. However, usually it is surface integrals that are of most concern, as with lift and drag. The first integrals, formal or analytic, will be real except if we say explicitly the converse. As before, many of these integrals have zero value. The first three integrals are also the first integrals for > 0 and the proof remains the same (see). These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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