Generalizations of pp-wave spacetimes in higher dimensions

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谱稀疏ppt

Spectral Compressive Sensing
• Compressive sensing applied to frequency-sparse signals
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linear measurements
nonzero DFT coefficients
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Structured Frequency-Sparse Signals
• If x is K-structured frequency-sparse, then there exists a K-sparse vector such that and the nonzeros in are spaced apart from each other.
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Signal is sum of 10 sinusoids
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[Blumensath, Davies; Lu, Do]
Leveraging Structure in Recovery
Many state-of-the-art sparse recovery algorithms (greedy and optimization solvers) rely on [Daubechies, Defrise, and DeMol; thresholding

论文翻译——超表面理论及应用

超表面理论及应用—超材料的平面化An Overview of the Theory and Applications of Metasurfaces: The Two-Dimensional Equivalents of MetamaterialsChristopher L。

Holloway1, Edward F. Kuester2, Joshua A。

Gordon1, John O'Hara3，Jim Booth1，and David R。

Smith4 三碗译摘要超材料通常由按一定规律排布的散射体或者通孔构成，由此来获得一定的性能指标。

这些期望的特性通常是天然材料所不具备的，比如负折射率和近零折射率等.在过去的十年里，超材料从理论概念走到了市场应用。

3D超材料也可以由二维表面来代替，也就是超表面,它是由很多小散射体或者孔组成的平面结构，在很多应用中，超表面可以达到超材料的效果。

超表面在占据的物理空间上比3D超材料有着优势，由此，超表面可以提供低耗能结构。

文章中将讨论到超表面特性的理论基础和它们不同的应用。

我们也将可以看出超表面和传统的频率选择表面的区别。

在电磁领域超表面有着很广泛的应用（从微波到可见光波段），包括智能控制表面、小型化的谐振腔、新型波导结构、角独立表面、吸收器、生物分子设备、THz调制和灵敏频率调节材料等等。

文中综述了近几年这种材料或者表面的发展，并让我们更加接近一百年前拉姆和Pocklington或者之后的Mandel和Veselago所提出的令人惊讶的观点.引言最近这些年，超材料这方面一直引领着材料的潮流。

超材料是一种新的人工合成材料来得到自然材料所不具备的一些特性。

在电磁背景中，这方面最早的实例就是人工电介质。

之后，我们将会看到和经典结构完全不同的超材料和超表面，比如光子能带隙结构（PBG)、频率选择表面(FSS）.双负指数（DNG)超材料是一种盛行的超材料，也叫作负指数材料(NIM）、左手材料等（LHM）。

Cubature Kalman Filters

1254IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009Cubature Kalman FiltersIenkaran Arasaratnam and Simon Haykin, Life Fellow, IEEEAbstract—In this paper, we present a new nonlinear ﬁlter for high-dimensional state estimation, which we have named the cubature Kalman ﬁlter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian ﬁlter. Speciﬁcally, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear ﬁltering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the ﬁrst problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear ﬁlters. Index Terms—Bayesian ﬁlters, cubature rules, Gaussian quadrature rules, invariant theory, Kalman ﬁlter, nonlinear ﬁltering.• Time update, which involves computing the predictive density(3)where denotes the history of input; is the measurement pairs up to time and the state transition old posterior density at time is obtained from (1). density • Measurement update, which involves computing the posterior density of the current stateI. INTRODUCTIONUsing the state-space model (1), (2) and Bayes’ rule we have (4) where the normalizing constant is given byIN this paper, we consider the ﬁltering problem of a nonlinear dynamic system with additive noise, whose statespace model is deﬁned by the pair of difference equations in discrete-time [1] (1) (2)is the state of the dynamic system at discrete where and are time ; is the known control input, some known functions; which may be derived from a compensator as in Fig. 1; is the measurement; and are independent process and measurement Gaussian noise sequences with zero and , respectively. means and covariances In the Bayesian ﬁltering paradigm, the posterior density of the state provides a complete statistical description of the state at that time. On the receipt of a new measurement at time , we in update the old posterior density of the state at time two basic steps:Manuscript received July 02, 2008; revised July 02, 2008, August 29, 2008, and September 16, 2008. First published May 27, 2009; current version published June 10, 2009. This work was supported by the Natural Sciences & Engineering Research Council (NSERC) of Canada. Recommended by Associate Editor S. Celikovsky. The authors are with the Cognitive Systems Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: aienkaran@grads.ece.mcmaster.ca; haykin@mcmaster. ca). Color versions of one or more of the ﬁgures in this paper are available online at . Digital Object Identiﬁer 10.1109/TAC.2009.2019800To develop a recursive relationship between the predictive density and the posterior density in (4), the inputs have to satisfy the relationshipwhich is also called the natural condition of control [2]. has sufﬁcient This condition therefore suggests that information to generate the input . To be speciﬁc, the can be generated using . Under this condiinput tion, we may equivalently write (5) Hence, substituting (5) into (4) yields (6) as desired, where (7) and the measurement likelihood function obtained from (2). is0018-9286/$25.00 © 2009 IEEEARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1255Fig. 1. Signal-ﬂow diagram of a dynamic state-space model driven by the feedback control input. The observer may employ a Bayesian ﬁlter. The label denotes the unit delay.The Bayesian ﬁlter solution given by (3), (6), and (7) provides a uniﬁed recursive approach for nonlinear ﬁltering problems, at least conceptually. From a practical perspective, however, we ﬁnd that the multi-dimensional integrals involved in (3) and (7) are typically intractable. Notable exceptions arise in the following restricted cases: 1) A linear-Gaussian dynamic system, the optimal solution for which is given by the celebrated Kalman ﬁlter [3]. 2) A discrete-valued state-space with a ﬁxed number of states, the optimal solution for which is given by the grid ﬁlter (Hidden-Markov model ﬁlter) [4]. 3) A “Benes type” of nonlinearity, the optimal solution for which is also tractable [5]. In general, when we are confronted with a nonlinear ﬁltering problem, we have to abandon the idea of seeking an optimal or analytical solution and be content with a suboptimal solution to the Bayesian ﬁlter [6]. In computational terms, suboptimal solutions to the posterior density can be obtained using one of two approximate approaches: 1) Local approach. Here, we derive nonlinear ﬁlters by ﬁxing the posterior density to take a priori form. For example, we may assume it to be Gaussian; the nonlinear ﬁlters, namely, the extended Kalman ﬁlter (EKF) [7], the central-difference Kalman ﬁlter (CDKF) [8], [9], the unscented Kalman ﬁlter (UKF) [10], and the quadrature Kalman ﬁlter (QKF) [11], [12], fall under this ﬁrst category. The emphasis on locality makes the design of the ﬁlter simple and fast to execute. 2) Global approach. Here, we do not make any explicit assumption about the posterior density form. For example, the point-mass ﬁlter using adaptive grids [13], the Gaussian mixture ﬁlter [14], and particle ﬁlters using Monte Carlo integrations with the importance sampling [15], [16] fall under this second category. Typically, the global methods suffer from enormous computational demands. Unfortunately, the presently known nonlinear ﬁlters mentioned above suffer from the curse of dimensionality [17] or divergence or both. The effect of curse of dimensionality may often become detrimental in high-dimensional state-space models with state-vectors of size 20 or more. The divergence may occur for several reasons including i) inaccurate or incomplete model of the underlying physical system, ii) informationloss in capturing the true evolving posterior density completely, e.g., a nonlinear ﬁlter designed under the Gaussian assumption may fail to capture the key features of a multi-modal posterior density, iii) high degree of nonlinearities in the equations that describe the state-space model, and iv) numerical errors. Indeed, each of the above-mentioned ﬁlters has its own domain of applicability and it is doubtful that a single ﬁlter exists that would be considered effective for a complete range of applications. For example, the EKF, which has been the method of choice for nonlinear ﬁltering problems in many practical applications for the last four decades, works well only in a ‘mild’ nonlinear environment owing to the ﬁrst-order Taylor series approximation for nonlinear functions. The motivation for this paper has been to derive a more accurate nonlinear ﬁlter that could be applied to solve a wide range (from low to high dimensions) of nonlinear ﬁltering problems. Here, we take the local approach to build a new ﬁlter, which we have named the cubature Kalman ﬁlter (CKF). It is known that the Bayesian ﬁlter is rendered tractable when all conditional densities are assumed to be Gaussian. In this case, the Bayesian ﬁlter solution reduces to computing multi-dimensional integrals, whose integrands are all of the form nonlinear function Gaussian. The CKF exploits the properties of highly efﬁcient numerical integration methods known as cubature rules for those multi-dimensional integrals [18]. With the cubature rules at our disposal, we may describe the underlying philosophy behind the derivation of the new ﬁlter as nonlinear ﬁltering through linear estimation theory, hence the name “cubature Kalman ﬁlter.” The CKF is numerically accurate and easily extendable to high-dimensional problems. The rest of the paper is organized as follows: Section II derives the Bayesian ﬁlter theory in the Gaussian domain. Section III describes numerical methods available for moment integrals encountered in the Bayesian ﬁlter. The cubature Kalman ﬁlter, using a third-degree spherical-radial cubature rule, is derived in Section IV. Our argument for choosing a third-degree rule is articulated in Section V. We go on to derive a square-root version of the CKF for improved numerical stability in Section VI. The existing sigma-point approach is compared with the cubature method in Section VII. We apply the CKF in two nonlinear state estimation problems in Section VIII. Section IX concludes the paper with a possible extension of the CKF algorithm for a more general setting.1256IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009II. BAYESIAN FILTER THEORY IN THE GAUSSIAN DOMAIN The key approximation taken to develop the Bayesian ﬁlter theory under the Gaussian domain is that the predictive density and the ﬁlter likelihood density are both Gaussian, which eventually leads to a Gaussian posterior den. The Gaussian is the most convenient and widely sity used density function for the following reasons: • It has many distinctive mathematical properties. — The Gaussian family is closed under linear transformation and conditioning. — Uncorrelated jointly Gaussian random variables are independent. • It approximates many physical random phenomena by virtue of the central limit theorem of probability theory (see Sections 5.7 and 6.7 in [19] for more details). Under the Gaussian approximation, the functional recursion of the Bayesian ﬁlter reduces to an algebraic recursion operating only on means and covariances of various conditional densities encountered in the time and the measurement updates. A. Time Update In the time update, the Bayesian ﬁlter computes the mean and the associated covariance of the Gaussian predictive density as follows: (8) where is the statistical expectation operator. Substituting (1) into (8) yieldsTABLE I KALMAN FILTERING FRAMEWORKB. Measurement Update It is well known that the errors in the predicted measurements are zero-mean white sequences [2], [20]. Under the assumption that these errors can be well approximated by the Gaussian, we write the ﬁlter likelihood density (12) where the predicted measurement (13) and the associated covariance(14) Hence, we write the conditional Gaussian density of the joint state and the measurement(15) (9) where the cross-covariance is assumed to be zero-mean and uncorrelated Because with the past measurements, we get (16) On the receipt of a new measurement , the Bayesian ﬁlter from (15) yielding computes the posterior density (17) (10) where is the conventional symbol for a Gaussian density. Similarly, we obtain the error covariance where (18) (19) (20) If and are linear functions of the state, the Bayesian ﬁlter under the Gaussian assumption reduces to the Kalman ﬁlter. Table I shows how quantities derived above are called in the Kalman ﬁltering framework. The signal-ﬂow diagram in Fig. 2 summarizes the steps involved in the recursion cycle of the Bayesian ﬁlter. The heart of the Bayesian ﬁlter is therefore how to compute Gaussian(11)ARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1257Fig. 2. Signal-ﬂow diagram of the recursive Bayesian ﬁlter under the Gaussian assumption, where “G-” stands for “Gaussian-.”weighted integrals whose integrands are all of the form nonGaussian density that are present in (10), linear function (11), (13), (14) and (16). The next section describes numerical integration methods to compute multi-dimensional weighted integrals. III. REVIEW ON NUMERICAL METHODS FOR MOMENT INTEGRALS Consider a multi-dimensional weighted integral of the form (21) is some arbitrary function, is the region of where for all integration, and the known weighting function . In a Gaussian-weighted integral, for example, is a Gaussian density and satisﬁes the nonnegativity condition in the entire region. If the solution to the above integral (21) is difﬁcult to obtain, we may seek numerical integration methods to compute it. The basic task of numerically computing the integral (21) is to ﬁnd a set of points and weights that approximates by a weighted sum of function evaluations the integral (22) The methods used to ﬁnd can be divided into product rules and non-product rules, as described next. A. Product Rules ), we For the simplest one-dimensional case (that is, may apply the quadrature rule to compute the integral (21) numerically [21], [22]. In the context of the Bayesian ﬁlter, we mention the Gauss-Hermite quadrature rule; when the is in the form of a Gaussian density weighting functionis well approximated by a polynomial and the integrand in , the Gauss-Hermite quadrature rule is used to compute the Gaussian-weighted integral efﬁciently [12]. The quadrature rule may be extended to compute multidimensional integrals by successively applying it in a tensorproduct of one-dimensional integrals. Consider an -point per dimension quadrature rule that is exact for polynomials of points for functional degree up to . We set up a grid of evaluations and numerically compute an -dimensional integral while retaining the accuracy for polynomials of degree up to only. Hence, the computational complexity of the product quadrature rule increases exponentially with , and therefore , suffers from the curse of dimensionality. Typically for the product Gauss-Hermite quadrature rule is not a reasonable choice to approximate a recursive optimal Bayesian ﬁlter. B. Non-Product Rules To mitigate the curse of dimensionality issue in the product rules, we may seek non-product rules for integrals of arbitrary dimensions by choosing points directly from the domain of integration [18], [23]. Some of the well-known non-product rules include randomized Monte Carlo methods [4], quasi-Monte Carlo methods [24], [25], lattice rules [26] and sparse grids [27]–[29]. The randomized Monte Carlo methods evaluate the integration using a set of equally-weighted sample points drawn randomly, whereas in quasi-Monte Carlo methods and lattice rules the points are generated from a unit hyper-cube region using deterministically deﬁned mechanisms. On the other hand, the sparse grids based on Smolyak formula in principle, combine a quadrature (univariate) routine for high-dimensional integrals more sophisticatedly; they detect important dimensions automatically and place more grid points there. Although the non-product methods mentioned here are powerful numerical integration tools to compute a given integral with a prescribed accuracy, they do suffer from the curse of dimensionality to certain extent [30].1258IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009C. Proposed Method In the recursive Bayesian estimation paradigm, we are interested in non-product rules that i) yield reasonable accuracy, ii) require small number of function evaluations, and iii) are easily extendable to arbitrarily high dimensions. In this paper we derive an efﬁcient non-product cubature rule for Gaussianweighted integrals. Speciﬁcally, we obtain a third-degree fullysymmetric cubature rule, whose complexity in terms of function evaluations increases linearly with the dimension . Typically, a set of cubature points and weights are chosen so that the cubature rule is exact for a set of monomials of degree or less, as shown by (23)Gaussian density. Speciﬁcally, we consider an integral of the form (24)deﬁned in the Cartesian coordinate system. To compute the above integral numerically we take the following two steps: i) We transform it into a more familiar spherical-radial integration form ii) subsequently, we propose a third-degree spherical-radial rule. A. Transformation In the spherical-radial transformation, the key step is a change of variable from the Cartesian vector to a radius and with , so direction vector as follows: Let for . Then the integral (24) can be that rewritten in a spherical-radial coordinate system as (25) is the surface of the sphere deﬁned by and is the spherical surface measure or the area element on . We may thus write the radial integral (26) is deﬁned by the spherical integral with the unit where weighting function (27) The spherical and the radial integrals are numerically computed by the spherical cubature rule (Section IV-B below) and the Gaussian quadrature rule (Section IV-C below), respectively. Before proceeding further, we introduce a number of notations and deﬁnitions when constructing such rules as follows: • A cubature rule is said to be fully symmetric if the following two conditions hold: implies , where is any point obtainable 1) from by permutations and/or sign changes of the coordinates of . on the region . That is, all points in 2) the fully symmetric set yield the same weight value. For example, in the one-dimensional space, a point in the fully symmetric set implies that and . • In a fully symmetric region, we call a point as a generator , where if , . The new should not be confused with the control input . zero coordinates and use • For brevity, we suppress to represent a complete fully the notation symmetric set of points that can be obtained by permutating and changing the sign of the generator in all possible ways. Of course, the complete set entails where; are non-negative integers and . Here, an important quality criterion of a cubature rule is its degree; the higher the degree of the cubature rule is, the more accurate solution it yields. To ﬁnd the unknowns of the cubature rule of degree , we solve a set of moment equations. However, solving the system of moment equations may be more tedious with increasing polynomial degree and/or dimension of the integration domain. For example, an -point cubature rule entails unknown parameters from its points and weights. In general, we may form a system of equations with respect to unknowns from distinct monomials of degree up to . For the nonlinear system to have at least one solution (in this case, the system is said to be consistent), we use at least as many unknowns as equations [31]. That is, we choose to be . Suppose we obtain a cu. In this case, we solve bature rule of degree three for nonlinear moment equations; the re) sulting rule may consist of more than 85 ( weighted cubature points. To reduce the size of the system of algebraically independent equations or equivalently the number of cubature points markedly, Sobolev proposed the invariant theory in 1962 [32] (see also [31] and the references therein for a recent account of the invariant theory). The invariant theory, in principle, discusses how to restrict the structure of a cubature rule by exploiting symmetries of the region of integration and the weighting function. For example, integration regions such as the unit hypercube, the unit hypersphere, and the unit simplex exhibit symmetry. Hence, it is reasonable to look for cubature rules sharing the same symmetry. For the case considered above and ), using the invariant theory, we may con( cubature points struct a cubature rule consisting of by solving only a pair of moment equations (see Section IV). Note that the points and weights of the cubature rule are in. Hence, they can be computed dependent of the integrand off-line and stored in advance to speed up the ﬁlter execution. where IV. CUBATURE KALMAN FILTER As described in Section II, nonlinear ﬁltering in the Gaussian domain reduces to a problem of how to compute integrals, whose integrands are all of the form nonlinear functionARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1259points when are all distinct. For example, represents the following set of points:Here, the generator is • We use . set B. Spherical Cubature Rule. to denote the -th point from theWe ﬁrst postulate a third-degree spherical cubature rule that takes the simplest structure due to the invariant theory (28) The point set due to is invariant under permutations and sign changes. For the above choice of the rule (28), the monomials with being an odd integer, are integrated exactly. In order that this rule is exact for all monomials of degree up to three, it remains to require that the rule is exact , 2. Equivalently, to for all monomials for which ﬁnd the unknown parameters and , it sufﬁces to consider , and due to the fully symmonomials metric cubature rule (29) (30) where the surface area of the unit sphere with . Solving (29) and (30) , and . Hence, the cubature points are yields located at the intersection of the unit sphere and its axes. C. Radial Rule We next propose a Gaussian quadrature for the radial integration. The Gaussian quadrature is known to be the most efﬁcient numerical method to compute a one-dimensional integration [21], [22]. An -point Gaussian quadrature is exact and constructed as up to polynomials of degree follows: (31) where is a known weighting function and non-negative on ; the points and the associated weights the interval are unknowns to be determined uniquely. In our case, a comparison of (26) and (31) yields the weighting function and and , respecthe interval to be tively. To transform this integral into an integral for which the solution is familiar, we make another change of variable via yielding. The integral on the right-hand side of where (32) is now in the form of the well-known generalized GaussLaguerre formula. The points and weights for the generalized Gauss-Laguerre quadrature are readily obtained as discussed elsewhere [21]. A ﬁrst-degree Gauss-Laguerre rule is exact for . Equivalently, the rule is exact for ; it . is not exact for odd degree polynomials such as Fortunately, when the radial-rule is combined with the spherical rule to compute the integral (24), the (combined) spherical-radial rule vanishes for all odd-degree polynomials; the reason is that the spherical rule vanishes by symmetry for any odd-degree polynomial (see (25)). Hence, the spherical-radial rule for (24) is exact for all odd degrees. Following this argument, for a spherical-radial rule to be exact for all third-degree polyno, it sufﬁces to consider the ﬁrst-degree genermials in alized Gauss-Laguerre rule entailing a single point and weight. We may thus write (33) where the point is chosen to be the square-root of the root of the ﬁrst-order generalized Laguerre polynomial, which is orthogonal with respect to the modiﬁed weighting function ; subsequently, we ﬁnd by solving the zeroth-order moment equation appropriately. In this case, we , and . A detailed account have of computing the points and weights of a Gaussian quadrature with the classical and nonclassical weighting function is presented in [33]. D. Spherical-Radial Rule In this subsection, we describe two useful results that are used to i) combine the spherical and radial rule obtained separately, and ii) extend the spherical-radial rule for a Gaussian weighted integral. The respective results are presented as two propositions: Proposition 4.1: Let the radial integral be computed numer-point Gaussian quadrature rule ically by theLet the spherical integral be computed numerically by the -point spherical ruleThen, an by-point spherical-radial cubature rule is given(34) Proof: Because cubature rules are devised to be exact for a subspace of monomials of some degree, we consider an integrand of the form(32)1260IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009where are some positive integers. Hence, we write the integral of interestwhereFor the moment, we assume the above integrand to be a mono. Making the mial of degree exactly; that is, change of variable as described in Section IV-A, we getWe use the cubature-point set to numerically compute integrals (10), (11), and (13)–(16) and obtain the CKF algorithm, details of which are presented in Appendix A. Note that the above cubature-point set is now deﬁned in the Cartesian coordinate system. V. IS THERE A NEED FOR HIGHER-DEGREE CUBATURE RULES? In this section, we emphasize the importance of third-degree cubature rules over higher-degree rules (degree more than three), when they are embedded into the cubature Kalman ﬁltering framework for the following reasons: • Sufﬁcient approximation. The CKF recursively propagates the ﬁrst two-order moments, namely, the mean and covariance of the state variable. A third-degree cubature rule is also constructed using up to the second-order moment. Moreover, a natural assumption for a nonlinearly transformed variable to be closed in the Gaussian domain is that the nonlinear function involved is reasonably smooth. In this case, it may be reasonable to assume that the given nonlinear function can be well-approximated by a quadratic function near the prior mean. Because the third-degree rule is exact up to third-degree polynomials, it computes the posterior mean accurately in this case. However, it computes the error covariance approximately; for the covariance estimate to be more accurate, a cubature rule is required to be exact at least up to a fourth degree polynomial. Nevertheless, a higher-degree rule will translate to higher accuracy only if the integrand is well-behaved in the sense of being approximated by a higher-degree polynomial, and the weighting function is known to be a Gaussian density exactly. In practice, these two requirements are hardly met. However, considering in the cubature Kalman ﬁltering framework, our experience with higher-degree rules has indicated that they yield no improvement or make the performance worse. • Efﬁcient and robust computation. The theoretical lower bound for the number of cubature points of a third-degree centrally symmetric cubature rule is given by twice the dimension of an integration region [34]. Hence, the proposed spherical-radial cubature rule is considered to be the most efﬁcient third-degree cubature rule. Because the number of points or function evaluations in the proposed cubature rules scales linearly with the dimension, it may be considered as a practical step for easing the curse of dimensionality. According to [35] and Section 1.5 in [18], a ‘good’ cubature rule has the following two properties: (i) all the cubature points lie inside the region of integration, and (ii) all the cubature weights are positive. The proposed rule equal, positive weights for an -dimensional entails unbounded region and hence belongs to a good cubature family. Of course, we hardly ﬁnd higher-degree cubature rules belonging to a good cubature family especially for high-dimensional integrations.Decomposing the above integration into the radial and spherical integrals yieldsApplying the numerical rules appropriately, we haveas desired. As we may extend the above results for monomials of degree less than , the proposition holds for any arbitrary integrand that can be written as a linear combination of monomials of degree up to (see also [18, Section 2.8]). Proposition 4.2: Let the weighting functions and be and . such that , we Then for every square matrix have (35) Proof: Consider the left-hand side of (35). Because a positive deﬁnite matrix, we factorize to be , we get Making a change of variable via is .which proves the proposition. For the third-degree spherical-radial rule, and . Hence, it entails a total of cubature points. Using the above propositions, we extend this third-degree spherical-radial rule to compute a standard Gaussian weighted integral as follows:ARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1261In the ﬁnal analysis, the use of higher-degree cubature rules in the design of the CKF may marginally improve its performance at the expense of a reduced numerical stability and an increased computational cost. VI. SQUARE-ROOT CUBATURE KALMAN FILTER This section addresses i) the rationale for why we need a square-root extension of the standard CKF and ii) how the square-root solution can be developed systematically. The two basic properties of an error covariance matrix are i) symmetry and ii) positive deﬁniteness. It is important that we preserve these two properties in each update cycle. The reason is that the use of a forced symmetry on the solution of the matrix Ricatti equation improves the numerical stability of the Kalman ﬁlter [36], whereas the underlying meaning of the covariance is embedded in the positive deﬁniteness. In practice, due to errors introduced by arithmetic operations performed on ﬁnite word-length digital computers, these two properties are often lost. Speciﬁcally, the loss of the positive deﬁniteness may probably be more hazardous as it stops the CKF to run continuously. In each update cycle of the CKF, we mention the following numerically sensitive operations that may catalyze to destroy the properties of the covariance: • Matrix square-rooting [see (38) and (43)]. • Matrix inversion [see (49)]. • Matrix squared-form amplifying roundoff errors [see (42), (47) and (48)]. • Substraction of the two positive deﬁnite matrices present in the covariant update [see (51)]. Moreover, some nonlinear ﬁltering problems may be numerically ill-conditioned. For example, the covariance is likely to turn out to be non-positive deﬁnite when i) very accurate measurements are processed, or ii) a linear combination of state vector components is known with greater accuracy while other combinations are essentially unobservable [37]. As a systematic solution to mitigate ill effects that may eventually lead to an unstable or even divergent behavior, the logical procedure is to go for a square-root version of the CKF, hereafter called square-root cubature Kalman ﬁlter (SCKF). The SCKF essentially propagates square-root factors of the predictive and posterior error covariances. Hence, we avoid matrix square-rooting operations. In addition, the SCKF offers the following beneﬁts [38]: • Preservation of symmetry and positive (semi)deﬁniteness of the covariance. Improved numerical accuracy owing to the fact that , where the symbol denotes the condition number. • Doubled-order precision. To develop the SCKF, we use (i) the least-squares method for the Kalman gain and (ii) matrix triangular factorizations or triangularizations (e.g., the QR decomposition) for covariance updates. The least-squares method avoids to compute a matrix inversion explicitly, whereas the triangularization essentially computes a triangular square-root factor of the covariance without square-rooting a squared-matrix form of the covariance. Appendix B presents the SCKF algorithm, where all of the steps can be deduced directly from the CKF except for the update of the posterior error covariance; hence we derive it in a squared-equivalent form of the covariance in the appendix.The computational complexity of the SCKF in terms of ﬂops, grows as the cube of the state dimension, hence it is comparable to that of the CKF or the EKF. We may reduce the complexity signiﬁcantly by (i) manipulating sparsity of the square-root covariance carefully and (ii) coding triangularization algorithms for distributed processor-memory architectures. VII. A COMPARISON OF UKF WITH CKF Similarly to the CKF, the unscented Kalman ﬁlter (UKF) is another approximate Bayesian ﬁlter built in the Gaussian domain, but uses a completely different set of deterministic weighted points [10], [39]. To elaborate the approach taken in the UKF, consider an -dimensional random variable having with mean and covariance a symmetric prior density , within which the Gaussian is a special case. Then a set of sample points and weights, are chosen to satisfy the following moment-matching conditions:Among many candidate sets, one symmetrically distributed sample point set, hereafter called the sigma-point set, is picked up as follows:where and the -th column of a matrix is denoted ; the parameter is used to scale the spread of sigma points by from the prior mean , hence the name “scaling parameter”. Due to its symmetry, the sigma-point set matches the skewness. Moreover, to capture the kurtosis of the prior density closely, it is sug(Appendix I of [10], gested that we choose to be [39]). This choice preserves moments up to the ﬁfth order exactly in the simple one-dimensional Gaussian case. In summary, the sigma-point set is chosen to capture a number as correctly as of low-order moments of the prior density possible. Then the unscented transformation is introduced as a method that are related to of computing posterior statistics of by a nonlinear transformation . It approximates the mean and the covariance of by a weighted sum of projected space, as shown by sigma points in the(36)(37)。

【doc】球面波分解理论及其倾斜叠加方法的实现

球面波分解理论及其倾斜叠加方法的实现7;一72000年12月石j由址球幽擢第35卷第6期球面波分解理论及其倾斜叠加方法的实现孙成禹(石油大学(东营)资源系).p引,年摘要孙成禹.球面波分解理论殛其倾斜叠加方法的实现.石油地球韧理勘探,2000,35(6):723~729本文首班对球面筒谐波和脉冲瘦的平面渡舟解进行了理论探讨,分析了现行r—P 变换方法及其与球面渡舟解的关系,提出了精确的球面脉冲渡记录舟解方法,并培出了数值算佣理论分析和数值试验结果表明,单纯的r-p变换不能真正实现对球面渡的平面玻舟解,而本文提出的方法则能较好地解袅球面瘦分解的问题.主题词倾斜叠::立塞篮球面渡平面渡波的分解ABSTRACT,面粑SunChengyu.Theoryofsphericalwavedecompositonandrealizationofdipstack.OGP,2000,35(6):723~729 Itisfirsttimepaperthattheplanarwavedecompositionbothforsphericalhat—monicwaveandfnrimpulsivewavehasbeentheoreticallydiscussedthearticleann—lyzedtherelationbetweenpresentr—Ptransformandspheriealwavedecomposition, putforwardaexactdecompositionmethodforsphericalimpulsiverecordsandgave anumericalexample.Theresultsoftheoreticalanalysisandnumericaltestshowthatsimpler—Ptransformcan'ttrulyrealizeaplanarwavedecompositionforspher—icalwave,themethodpresentedinthepapercanbetterresolvetheproblemof sphericalwavedecomposition.Subjectterms:dipstack,rPtransform,sphericalwave,planarwave,wavede—composition引言目前大多数波动理论和实用技术是建立在平面渡假设基础上,而实际中广泛应用的却是点震源.由于理论上的复杂性,完全建立在球面波模型上的地震处理和解释方法尚未形成.因此,将球面波记录先分解成平面波记录[,再利用平面波理论进行分析处理,具有一定的实际意义.球面波可以表示成不同平面波的积分形式':;反之,也可将球面波分解成各种平面渡SunCheng~,DepartmentofPetroleumR~sourceSciencetPetroleumUniversityofChina, Dong~dngCity,ShandongProvince,257061,China本文于1999年i0月18日收到72I石油地球物理勘探2000年分量.Brysk等(1986)0],刘清林等(1988)一均认为利用柱坐标下的倾斜叠加(或称为柱面户变换)就可直接实现球面波的平面波分解,前者还给出了建立在简谐波基础上的具体的数值算法.Wang等(1989)口一提出利用倾斜叠加可以完成地面地震资料的球面扩散校正,并给出了墨西哥湾的处理实例.李云典等(1994)_6实现了r—P域的A VO分析.Stoffa(1989),Wang等(1989)都认为}户域资料也适用于滤波,层速度分析,AVO分析,亮点分析,速度和密度反演及其它在时空域能进行的处理和分析工作.柱坐标倾斜叠加是否真正可以将球面渡分解为能与原波场相对比的平面渡,这需要从波形特征和振幅特征等方面进行检验.我们通过理论上的严密推导,提出了自己的球面脉冲波分解方法,并利用数值算例作了对比分析,得到了几点结论.球面简谐波和脉冲波的平面波分解球面简谐波的分解关于简谐球面波的合成理论,已有多人给出过解答(Ewing等,1957):.其中,使用较为广泛的是Sommerfeld积分],[e一RR(1)式中:一f:一筹1寺一∞(p'--c)音;为沿方向传播的平面波视波数;为球面波波速;对于1均匀平面波,户(<f叫)为沿r方向的视慢度,也称射线参数;l,.(∞户r)为零阶Bessel函数;素为球面扩散因子,满足R一++z.一r+.波传相对几何关系见图1.图1渡传几何关系示意图衰减的不均匀平面波.考虑球面波的振幅特征和时延性,式(1)可写成一A(e (2)令f=(一Pz){,则—.对Sommerfeld积分而言,要求Re{)≥0,因此这里Im{f)≤O.代入上式,可得Ae(t一譬RAJ.(小…e(3)由式(3)推知:球面波芸是由一系列平面波Ae加权叠加而成的,加权系数为一詈('r).当Im{f)=0时为均匀平面波,当Im{f)<O时为振幅随传播而以图1中上部区域为例,显然z>0,由式(3)可得j—f瓦A(f一)式(4)实现了简谐球面波的平面波分解(4)第35卷第6期孙成禹:球面渡分解理论及其倾斜叠加方法的实现725球面脉冲波的分解对于脉冲波,令式(4)右端被积函数中的簧一去d为球面脉冲波离开震源R处的频谱,对其积分,则有=[幽㈣一』:1一小则(,R)为对应的球面脉冲波.再设()=_..5(山)ed(7)与传播距离R无关,它表示一平面脉冲波,且在震源附近与上述球面波具有相同的强度(频谱相同).称为等源振幅波.由式(6)得』:e—d=jo~S()(8)则与式(4)对应的脉冲波分解方程为):一d.(~opr)rdr(9)≠0)=fl—上_—.(9)当m=.时'.一表示沿轴方向视速度为专一1=的均匀平面波(是R与.轴方向的夹角,图1),它具有与参与分解的球面脉冲波在震源附近相同的振幅频率特征;当Im{f1<O时,则可表示成≠0缸)去J一5(∞)e～edm的形式,它是一不均匀平面波(n为衰减函数).综合上述积分,式(9)可以表示成如下形式一一.』:()兰.,即通过三次积分变换,完成了球面脉冲波向平面波的分解r变换与球面波分解地面地震记录的一维r一户正变换公式为r,户)一f,,,.=.)d一』:(r+肛,.1y.z—o)如(11)石油地球物理勘探2000妊一般认为,一维r—P变换只适用于线震源情况,它可以将线源记录(柱面波)分解成平面波分量.而对于点震源记录,最好应使用如下的二维v-p变换公式来实现球面波的分解～[.≠2(r,户)一ldxl(r+P+PY,,Y,2—0)dy(12)Brysk和McCowan(1986)将它变换到柱坐标系下.设地层是轴对称的,即在各个方向上具有相同的性质,通过坐标变换~(reosg,rsP(peosa,psina),【(,)一,化简整理后,式(12)可写成(r,户)一2ld户(户一P耻)专Irdr[u(r+Pr,r,0,o)+u(r—Pr,r,0,0)](14)该式被称为柱面}户变换或柱面倾斜叠加公式.Brysk和McCowan(1986)将上述内积分看成对地震记录按炮检距进行加权后的一维r-p变换,变换后相当于得到了线源记录;而外积分则是对线源记录的柱面倾斜叠加,可得到平面波记录;并给出了计算外积分的具体数值算法.实际上,将式(13)代人式(12),整理可得(r,P,)一lrld≠"[r—prcos(≠一),r,≠,0]一IrdrId≠Idt{[r+prcos(≠一),r,≠,o]a[t—r—prcos(≠一.)]}(15)利用6函数的Fourier变换,式(15)可写成(r,户,)一,drr_ddtu[r+preos(≠一).r,≠,o]x×Iexp{j-[t—f—prcos(~一)]}(16)在轴对称简化的前提下,波场与方位无关.变化积分次序,式(16)中对≠的积分部分为Bessel函数积分式,即一d~xp[一jwpreos(≯一)]=(wpr)(17)代人式(16),则二维v-p变换公式即为≠(r,户)=ld∽lrd^,0()ldfe(,r)(18)相当于在频率域对地面地震记录作了一次Fourier—Bessel积分变换(见文献3的公式(2)).对比并分析式(18)和式(10)可知:(1)球面脉冲波分解(SID,式(10))是对地面记录的时间导数进行叠加;而柱面P变换(CSS,式(18))则是直接对记录波场进行叠加.(2)对于分解所得的平面波振幅,球面波分解在积分变换完成后,按其传播方向以因子f一(f～一p2)作加权处理;而柱面v-p变换则没有这种处理.(3)由图1可以看出,在.方向视速度为f一的平面波,在垂直方向上同沿R方向以速度c传播的平面波具有相同的走时特征{而柱面r—P变换则没表现出这种特征.第35卷第6期孙成禹:球面渡分解理论厦其倾斜叠加方法的宴现727因此,从理论上不能说柱面r—变换就是球面渡分解,这两者并不相同.利用Brysk 等(1986)的柱面r—变换子程序,球面脉冲渡分解(SID)的实际算法可以表示为≠(r,p):~cssl"](19)LuJ式中CSS表示进行柱面倾斜叠加可利用式(14)提供的数值算法.由于f本身也是p 的函数,,p'.;一r为垂直旅行时,故这里将变量统一写成(r.).时延因子ft一l表示从震源(反射地震l'f记录中为震源影像点)传到观测点的时间,等同于地面记录"(,r,0,0).数值算例及对比分析借助Ostrander(1984):建立的三层含气模型(图2),用主频为35Hz的Ricker子渡合成了共中心点平面渡反射记录,其反射渡振幅可用Zoeppritz方程精确求出(图3).考虑到实际生产中地面地震记录的观测方式和覆盖次数,我们取道间距为dx一25m,共合成了75道记录.该合成数据中不存在球面渡的渡前几何扩散因素;反射振幅A VO特征明显.利用同一模型参数,又合成了球面波反射记录(图4).由于是三层介质,且为非法向入射,为保证图2模型舟质参数V为缴被速度(m/s);P为密度(g!cm){d为泊捂比合成的球面波记录真正具有其应有的球面波振幅和波形特征,根据Huygens原理和衍射渡叠加理论,使用Kirchhoff衍射公式计算反射球面波渡场"cz,,.,一aI—]cz.其中为从源点到衍射点的距离;为从衍射点刊观测点的距离;,(￡)为人射渡;巩和口分别为rc和r与铅直方向的夹角;c为波速;f为反射系数.其道间距和总道数同平面渡情形.O2O04006∞800图3Ostrander模型平面渡台成记录(一域)图4Ostrander模型球面渡合成记录(z0域)图5为利用球面脉冲渡分解方法(SID)得到的p域记录,图8为利用柱面倾斜叠加方法(CSS)得到的域记录根据围3和图4数据中最大炮检距所对应的最大地面视速度,确定图5和图8中各户道间隔为dp=3×10～,也合成了75道.728石油地球物理勘探2000正图5球面波记录SID处理结果(rp域)图6球面波记录CSS处理结果(f—P域)可以从振幅和波形两方面来进行对比.图7中曲线Aa～曲线Dd分别给出了图3～图6各图中两个反射渡振幅随炮检距(或水平慢度P)的变化曲线(其中小写字母为第一个反射渡的负振幅曲线,大写字母为第二个反射波的正振幅曲线).曲线Aa是平面波振幅曲线,曲线Bb是球面波振幅曲线.显然,曲线C与曲线A最接近,说明SID法所得图5中正反射振幅恢复得最好.对于负反射,曲线c 与曲线a在小炮检距(或小)处接近,而在大炮检距(或大P)处曲线c,d与曲线a的相似性都不太好.其原因可能是由于浅层远道入射角大,模型数据(图4)制作不够准确;同时,由于空间域的-z道和rP域的P道不完全是一一对应的,因此振幅变化趋势可不完全一样.而用CSS 方法所得的曲线Dd离曲线Aa较远,说明振幅恢复效果较振幅O≥蔓三二歹…...一一…::::二/—————～———一20406o道号兰........～～:二:≥:≥,.一一.图7图3～图6各剖面振幅变化曲线第35卷第6期孙成禹:球面被分解理论及其倾斜叠加方法的实现729差.曲线C在前半段有一些跳动,主要是由于变换时的截断效应和空间假频引人的干扰所致.这类问题可通过加时窗等方法得到较好的解决.再对比变换前后的波形.随机抽取图3～图6中第12道,对应图8a～图8d.可看出用SID方法得到的波形(图8c)同模型的原始波形(图8a,图8b)一样,证明了SID方法的正确性;而用CSS方法得到的波形(图8d)却有较大改变,从零相位变成为最小相位,若用这样的剖面进行标定或作其它处理,就会引起较大误差.图8c和图8d中出现的一些小干扰也是■—-.-——,{=i,,一-.～'(a)(b]((d]图8图3～图6中左起第12道披形对比由两次积分变换时的截断效应和假频造成的.另外,由于在两个域中的时距曲线不同,因此图8a和图8b显示的渡的初至时间与图8c,图8d的略有不同.结论通过上述理论分析和数值试验,得到如下结论:(1)柱面倾斜叠加本身并不能将球面波分解成平面波.因此,在对地面地震记录简单地作r-p变换后,在r-p域内进行的一系列处理和分析都是不可靠的.(2)本文提出的球面脉冲渡分解方法(SID)能够较好地完成球面渡的平面波分解,并可针对分解变换后的平面波记录作进一步的理论分析和实例数值演算{在试验成功的基础上发展和完善了r—P域的处理和解释技术.参考文献1AkiK,RichardsPG.~mitativeseismo~ogy,V ol1{Theoryandmethod,wHFreemanandCo mpany.19802EwingWM,JardetzkyWS,PressF.Elastic口inlayeredmedia.McGraw—Hillbookc0mpany.NewY ork.19573BryskH,McCowanDW.Aslantstackprocedureforpoint—sourcedata. Geophysics,1986,51(7);1370~13864WangDY,McCowanDW.Sphericaldivergencecorrectionforseismicdatausingslantstack s,G.一physics,1989.54(5).563～5695刘清林,何樵登-Tau一一变换与nu域偏移6李云典?孙成禹,曲良河等.r一声域AVO分析7StoffaPL着,征廷璋译.Tau一声:另一种滤披工业出版社.1991:857~861石油地球物理勘探,1988,23(2):17l～187石油地球物理勘探,1994.89(4):413~422速度分析和成像域.SEG第59届年会论文集(1989).石油8OstranderWJ,Planewavereflectioncoefficientsfogassandatnorrealanglesofincidence. Geophysic.1984,49(1O):1637～16489江则荣,姜绍仁,夏戡原-Tau一变换若干问题的讨论及其在反演声纳浮标资料中的应用.石油地球物理勘探,1989,24(2):13O～143(本文编辑:来汉东)。

SCI写作句型汇总

S C I论文写作中一些常用的句型总结(一)很多文献已经讨论过了一、在Introduction里面经常会使用到的一个句子：很多文献已经讨论过了。

它的可能的说法有很多很多，这里列举几种我很久以前搜集的：A.??Solar energy conversion by photoelectrochemical cells?has been intensively investigated.?(Nature 1991, 353, 737 - 740?)B.?This was demonstrated in a number of studies that?showed that composite plasmonic-metal/semiconductor photocatalysts achieved significantly higher rates in various photocatalytic reactions compared with their pure semiconductor counterparts.C.?Several excellent reviews describing?these applications are available, and we do not discuss these topicsD.?Much work so far has focused on?wide band gap semiconductors for water splitting for the sake of chemical stability.（DOI:10.1038/NMAT3151）E.?Recent developments of?Lewis acids and water-soluble organometalliccatalysts?have attracted much attention.（Chem. Rev. 2002, 102, 3641?3666）F.?An interesting approach?in the use of zeolite as a water-tolerant solid acid?was described by?Ogawa et al（Chem.Rev. 2002, 102, 3641?3666）G.?Considerable research efforts have been devoted to?the direct transition metal-catalyzed conversion of aryl halides toaryl nitriles. (J. Org. Chem. 2000, 65, 7984-7989) H.?There are many excellent reviews in the literature dealing with the basic concepts of?the photocatalytic processand the reader is referred in particular to those by Hoffmann and coworkers,Mills and coworkers, and Kamat.(Metal oxide catalysis,19,P755)I. Nishimiya and Tsutsumi?have reported on（proposed）the influence of the Si/Al ratio of various zeolites on the acid strength, which were estimated by calorimetry using ammonia. (Chem.Rev. 2002, 102, 3641?3666)二、在results and discussion中经常会用到的：如图所示A. GIXRD patterns in?Figure 1A show?the bulk structural information on as-deposited films.?B.?As shown in Figure 7B,?the steady-state current density decreases after cycling between 0.35 and 0.7 V, which is probably due to the dissolution of FeOx.?C.?As can be seen from?parts a and b of Figure 7, the reaction cycles start with the thermodynamically most favorable VOx structures（J. Phys. Chem. C 2014, 118, 24950?24958）这与XX能够相互印证：A.?This is supported by?the appearance in the Ni-doped compounds of an ultraviolet–visible absorption band at 420–520nm (see Fig. 3 inset), corresponding to an energy range of about 2.9 to 2.3 eV.B. ?This?is consistent with the observation from?SEM–EDS. (Z.Zou et al. / Chemical Physics Letters 332 (2000) 271–277)C.?This indicates a good agreement between?the observed and calculated intensities in monoclinic with space groupP2/c when the O atoms are included in the model.D. The results?are in good consistent with?the observed photocatalytic activity...E. Identical conclusions were obtained in studies?where the SPR intensity and wavelength were modulated by manipulating the composition, shape,or size of plasmonic nanostructures.?F.??It was also found that areas of persistent divergent surfaceflow?coincide?with?regions where convection appears to be consistently suppressed even when SSTs are above 27.5°C.(二)1. 值得注意的是...A.?It must also be mentioned that?the recycling of aqueous organic solvent is less desirable than that of pure organic liquid.B.?Another interesting finding is that?zeolites with 10-membered ring pores showed high selectivities (>99%) to cyclohexanol, whereas those with 12-membered ring pores, such as mordenite, produced large amounts of dicyclohexyl ether. (Chem. Rev. 2002, 102,3641?3666)C.?It should be pointed out that?the nanometer-scale distribution of electrocatalyst centers on the electrode surface is also a predominant factor for high ORR electrocatalytic activity.D.?Notably,?the Ru II and Rh I complexes possessing the same BINAP chirality form antipodal amino acids as the predominant products.?(Angew. Chem. Int. Ed., 2002, 41: 2008–2022)E. Given the multitude of various transformations published,?it is noteworthy that?only very few distinct?activation?methods have been identified.?(Chem. Soc. Rev., 2009,?38, 2178-2189)F.?It is important to highlight that?these two directing effects will lead to different enantiomers of the products even if both the “H-bond-catalyst” and the?catalyst?acting by steric shielding have the same absolute stereochemistry. (Chem. Soc. Rev.,?2009,?38, 2178-2189)G.?It is worthwhile mentioning that?these PPNDs can be very stable for several months without the observations of any floating or precipitated dots, which is attributed to the electrostatic repulsions between the positively charge PPNDs resulting in electrosteric stabilization.(Adv. Mater., 2012, 24: 2037–2041)2.?...仍然是个挑战A.?There is thereby an urgent need but it is still a significant challenge to?rationally design and delicately tail or the electroactive MTMOs for advanced LIBs, ECs, MOBs, and FCs.?（Angew. Chem. Int. Ed.2 014, 53, 1488 – 1504）B.?However, systems that are?sufficiently stable and efficient for practical use?have not yet been realized.C.??It?remains?challenging?to?develop highly active HER catalysts based on materials that are more abundant at lower costs. (J. Am. Chem.Soc.,?2011,?133, ?7296–7299)D.?One of the?great?challenges?in the twenty-first century?is?unquestionably energy storage. (Nature Materials?2005, 4, 366 - 377?)众所周知A.?It is well established (accepted) / It is known to all / It is commonlyknown?that?many characteristics of functional materials, such as composition, crystalline phase, structural and morphological features, and the sur-/interface properties between the electrode and electrolyte, would greatly influence the performance of these unique MTMOs in electrochemical energy storage/conversion applications.（Angew. Chem. Int. Ed.2014,53, 1488 – 1504）B.?It is generally accepted (believed) that?for a-Fe2O3-based sensors the change in resistance is mainly caused by the adsorption and desorption of gases on the surface of the sensor structure. (Adv. Mater. 2005, 17, 582)C.?As we all know,?soybean abounds with carbon,?nitrogen?and oxygen elements owing to the existence of sugar,?proteins?and?lipids. (Chem. Commun., 2012,?48, 9367-9369)D.?There is no denying that?their presence may mediate spin moments to align parallel without acting alone to show d0-FM. (Nanoscale, 2013,?5, 3918-3930)(三)1. 正如下文将提到的...A.?As will be described below（也可以是As we shall see below）,?as the Si/Al ratio increases, the surface of the zeolite becomes more hydrophobic and possesses stronger affinity for ethyl acetate and the number of acid sites decreases.（Chem. Rev. 2002, 102, 3641?3666）B. This behavior is to be expected and?will?be?further?discussed?below. (J. Am. Chem. Soc.,?1955,?77, 3701–3707)C.?There are also some small deviations with respect to the flow direction,?whichwe?will?discuss?below.（Science, 2001, 291, 630-633）D.?Below,?we?will?see?what this implies.E.?Complete details of this case?will?be provided at a?later?time.E.?很多论文中，也经常直接用see below来表示，比如：The observation of nanocluster spheres at the ends of the nanowires is suggestive of a VLS growth process (see?below). (Science, 1998, ?279, 208-211)2. 这与XX能够相互印证...A.?This is supported by?the appearance in the Ni-doped compounds of an ultraviolet–visible absorption band at 420–520 nm (see Fig. 3 inset), corresponding to an energy range of about 2.9 to 2.3 eVB.This is consistent with the observation from?SEM–EDS. (Chem. Phys. Lett. 2000, 332, 271–277)C.?Identical conclusions were obtained?in studies where the SPR intensity and wavelength were modulated by manipulating the composition, shape, or size of plasmonic nanostructures.?（Nat. Mater. 2011, DOI: 10.1038/NMAT3151）D. In addition, the shape of the titration curve versus the PPi/1 ratio,?coinciding withthat?obtained by fluorescent titration studies, suggested that both 2:1 and 1:1 host-to-guest complexes are formed. (J. Am. Chem. Soc. 1999, 121, 9463-9464)E.?This unusual luminescence behavior is?in accord with?a recent theoretical prediction; MoS2, an indirect bandgap material in its bulk form, becomes a direct bandgapsemiconductor when thinned to a monolayer.?(Nano Lett.,?2010,?10, 1271–1275)3.?我们的研究可能在哪些方面得到应用A.?Our ?ndings suggest that?the use of solar energy for photocatalytic watersplitting?might provide a viable source for?‘clean’ hydrogen fuel, once the catalyticef?ciency of the semiconductor system has been improved by increasing its surface area and suitable modi?cations of the surface sites.B. Along with this green and cost-effective protocol of synthesis,?we expect that?these novel carbon nanodots?have potential applications in?bioimaging andelectrocatalysis.(Chem. Commun., 2012,?48, 9367-9369)C.?This system could potentially be applied as?the gain medium of solid-state organic-based lasers or as a component of high value photovoltaic (PV) materials, where destructive high energy UV radiation would be converted to useful low energy NIR radiation. (Chem. Soc. Rev., 2013,?42, 29-43)D.?Since the use of?graphene?may enhance the photocatalytic properties of TiO2?under UV and visible-light irradiation,?graphene–TiO2?composites?may potentially be usedto?enhance the bactericidal activity.?(Chem. Soc. Rev., 2012,?41, 782-796)E.??It is the first report that CQDs are both amino-functionalized and highly fluorescent,?which suggests their promising applications in?chemical sensing.(Carbon, 2012,?50,?2810–2815)(四)1. 什么东西还尚未发现/系统研究A. However,systems that are sufficiently stable and efficient for practical use?have not yet been realized.B. Nevertheless,for conventional nanostructured MTMOs as mentioned above,?some problematic disadvantages cannot be overlooked.（Angew. Chem. Int. Ed.2014,53, 1488 – 1504）C.?There are relatively few studies devoted to?determination of cmc values for block copolymer micelles. (Macromolecules 1991, 24, 1033-1040)D. This might be the reason why, despite of the great influence of the preparation on the catalytic activity of gold catalysts,?no systematic study concerning?the synthesis conditions?has been published yet.?(Applied Catalysis A: General2002, 226, ?1–13)E.?These possibilities remain to be?explored.F.??Further effort is required to?understand and better control the parameters dominating the particle surface passivation and resulting properties for carbon dots of brighter photoluminescence. (J. Am. Chem. Soc.,?2006,?128?, 7756–7757)2.?由于/因为...A.?Liquid ammonia?is particularly attractive as?an alternative to water?due to?its stability in the presence of strong reducing agents such as alkali metals that are used to access lower oxidation states.B.?The unique nature of?the cyanide ligand?results from?its ability to act both as a σdonor and a π acceptor combined with its negativecharge and ambidentate nature.C.?Qdots are also excellent probes for two-photon confocalmicroscopy?because?they are characterized by a very large absorption cross section?(Science ?2005,?307, 538-544).D.?As a result of?the reductive strategy we used and of the strong bonding between the surface and the aryl groups, low residual currents (similar to those observed at a bare electrode) were obtained over a large window of potentials, the same as for the unmodified parent GC electrode. (J. Am. Chem. Soc. 1992, 114, 5883-5884)E.?The small Tafel slope of the defect-rich MoS2 ultrathin nanosheets is advantageous for practical?applications,?since?it will lead to a faster increment of HER rate with increasing overpotential.(Adv. Mater., 2013, 25: 5807–5813)F. Fluorescent carbon-based materials have drawn increasing attention in recent years?owing to?exceptional advantages such as high optical absorptivity, chemical stability, biocompatibility, and low toxicity.(Angew. Chem. Int. Ed., 2013, 52: 3953–3957)G.??On the basis of?measurements of the heat of immersion of water on zeolites, Tsutsumi etal. claimed that the surface consists of siloxane bondings and is hydrophobicin the region of low Al content. (Chem. Rev. 2002, 102, 3641?3666)H.?Nanoparticle spatial distributions might have a large significance for catalyst stability,?given that?metal particle growth is a relevant deactivation mechanism for commercial catalysts.?3. ...很重要A.?The inhibition of additional nucleation during growth, in other words, the complete separation?of nucleation and growth,?is?critical(essential, important)?for?the successful synthesis of monodisperse nanocrystals. (Nature Materials?3, 891 - 895 (2004))B.??In the current study,?Cys,?homocysteine?(Hcy) and?glutathione?(GSH) were chosen as model?thiol?compounds since they?play important (significant, vital, critical) roles?in many biological processes and monitoring of these?thiol?compounds?is of great importance for?diagnosis of diseases.(Chem. Commun., 2012,?48, 1147-1149)C.?This is because according to nucleation theory,?what really matters?in addition to the change in temperature ΔT?(or supersaturation) is the cooling rate.(Chem. Soc. Rev., 2014,?43, 2013-2026)(五)1. 相反/不同于A.?On the contrary,?mononuclear complexes, called single-ion magnets (SIM), have shown hysteresis loops of butterfly/phonon bottleneck type, with negligiblecoercivity, and therefore with much shorter relaxation times of magnetization. (Angew. Chem. Int. Ed., 2014, 53: 4413–4417)B.?In contrast,?the Dy compound has significantly larger value of the transversal magnetic moment already in the ground state (ca. 10?1?μB), therefore allowing a fast QTM. （Angew. Chem. Int. Ed., 2014, 53: 4413–4417）C.?In contrast to?the structural similarity of these complexes, their magnetic behavior exhibits strong divergence.?(Angew. Chem. Int. Ed., 2014, 53: 4413–4417)D.?Contrary to?other conducting polymer semiconductors, carbon nitride ischemically and thermally stable and does not rely on complicated device manufacturing. (Nature materials, 2009, 8(1): 76-80.)E.?Unlike?the spherical particles they are derived from that Rayleigh light-scatter in the blue, these nanoprisms exhibit scattering in the red, which could be useful in developing multicolor diagnostic labels on the basis not only of nanoparticle composition and size but also of shape. (Science 2001,? 294, 1901-1903)2. 发现，阐明，报道，证实可供选择的词包括：verify, confirm, elucidate, identify, define, characterize, clarify, establish, ascertain, explain, observe, illuminate, illustrate，demonstrate, show, indicate, exhibit, presented, reveal, display, manifest，suggest, propose, estimate, prove, imply, disclose，report, describe，facilitate the identification of?举例：A. These stacks appear as nanorods in the two-dimensional TEM images, but tilting experiments?confirm that they are nanoprisms.?(Science 2001,? 294, 1901-1903)B. Note that TEM?shows?that about 20% of the nanoprisms are truncated.?(Science 2001,? 294, 1901-1903)C. Therefore, these calculations not only allow us to?identify?the important features in the spectrum of the nanoprisms but also the subtle relation between particle shape and the frequency of the bands that make up their spectra.?(Science 2001,? 294, 1901-1903)D. We?observed?a decrease in intensity of the characteristic surface plasmon band in the ultraviolet-visible (UV-Vis) spectroscopy for the spherical particles at λmax?= 400 nm with a concomitant growth of three new bands of λmax?= 335 (weak), 470 (medium), and 670 nm (strong), respectively. (Science 2001,? 294, 1901-1903)E. In this article, we present data?demonstrating?that opiate and nonopiate analgesia systems can be selectively activated by different environmental manipulationsand?describe?the neural circuitry involved. (Science 1982, 216, 1185-1192)F. This?suggests?that the cobalt in CoP has a partial positive charge (δ+), while the phosphorus has a partial negative charge (δ?),?implying?a transfer of electron density from Co to P.?(Angew. Chem., 2014, 126: 6828–6832)3. 如何指出当前研究的不足A. Although these inorganic substructures can exhibit a high density of functional groups, such as bridging OH groups, and the substructures contribute significantly to the adsorption properties of the material,surprisingly little attention has been devoted to?the post-synthetic functionalization of the inorganic units within MOFs. (Chem. Eur. J., 2013, 19: 5533–5536.)B.?Little is known,?however, about the microstructure of this material. (Nature Materials 2013,12, 554–561)C.?So far, very little information is available, and only in?the absorber film, not in the whole operational devices. （Nano Lett.,?2014,?14?(2), pp 888–893）D.?In fact it should be noted that very little optimisation work has been carried out on?these devices. (Chem. Commun., 2013,?49, 7893-7895)E. By far the most architectures have been prepared using a solution processed perovskite material,?yet a few examples have been reported that?have used an evaporated perovskite layer. (Adv. Mater., 2014, 27: 1837–1841.)F. Water balance issues have been effectively addressed in PEMFC technology through a large body of work encompassing imaging, detailed water content and water balance measurements, materials optimization and modeling,?but very few of these activities have been undertaken for?anion exchange membrane fuel cells,? primarily due to limited materials availability and device lifetime. (J. Polym. Sci. Part B: Polym. Phys., 2013, 51: 1727–1735)G. However,?none of these studies?tested for Th17 memory, a recently identified T cell that specializes in controlling extracellular bacterial infections at mucosal surfaces. (PNAS, 2013,?111, 787–792)H. However,?uncertainty still remains as to?the mechanism by which Li salt addition results in an extension of the cathodic reduction limit. (Energy Environ. Sci., 2014,?7, 232-250)I.?There have been a number of high profile cases where failure to?identify the most stable crystal form of a drug has led to severe formulation problems in manufacture. (Chem. Soc. Rev., 2014,?43, 2080-2088)J. However,?these measurements systematically underestimate?the amount of ordered material. ( Nature Materials 2013, 12, 1038–1044)(六)1.?取决于a.?This is an important distinction, as the overall activity of a catalyst will?depend on?the material properties, synthesis method, and other possible species that can be formed during activation.?(Nat. Mater.?2017,16,225–229)b.?This quantitative partitioning?was determined by?growing crystals of the 1:1 host–guest complex between?ExBox4+?and corannulene. (Nat. Chem.?2014,?6177–178)c.?They suggested that the Au particle size may?be the decisive factor for?achieving highly active Au catalysts.(Acc. Chem. Res.,?2014,?47, 740–749)d.?Low-valent late transition-metal catalysis has?become indispensable to?chemical synthesis, but homogeneous high-valent transition-metal catalysis is underdeveloped, mainly owing to the reactivity of high-valent transition-metal complexes and the challenges associated with synthesizing them.?(Nature2015,?517,449–454)e.?The polar effect?is a remarkable property that enables?considerably endergonic C–H abstractions?that would not be possible otherwise.?(Nature?2015, 525, 87–90)f.?Advances in heterogeneous catalysis?must rely on?the rational design of new catalysts. (Nat. Nanotechnol.?2017, 12, 100–101)g.?Likely, the origin of the chemoselectivity may?be also closely related to?the H?bonding with the N or O?atom of the nitroso moiety, a similar H-bonding effect is known in enamine-based nitroso chemistry. (Angew. Chem. Int. Ed.?2014, 53: 4149–4153)2.?有很大潜力a.?The quest for new methodologies to assemble complex organic molecules?continues to be a great impetus to?research efforts to discover or to optimize new catalytic transformations. (Nat. Chem.?2015,?7, 477–482)b.?Nanosized faujasite (FAU) crystals?have great potential as?catalysts or adsorbents to more efficiently process present and forthcoming synthetic and renewablefeedstocks in oil refining, petrochemistry and fine chemistry. (Nat. Mater.?2015, 14, 447–451)c.?For this purpose, vibrational spectroscopy?has proved promising?and very useful.?(Acc Chem Res. 2015, 48, 407–413.)d.?While a detailed mechanism remains to be elucidated and?there is room for improvement?in the yields and selectivities, it should be remarked that chirality transfer upon trifluoromethylation of enantioenriched allylsilanes was shown. (Top Catal.?2014,?57: 967.?)e.?The future looks bright for?the use of PGMs as catalysts, both on laboratory and industrial scales, because the preparation of most kinds of single-atom metal catalyst is likely to be straightforward, and because characterization of such catalysts has become easier with the advent of techniques that readily discriminate single atoms from small clusters and nanoparticles. (Nature?2015, 525, 325–326)f.?The unique mesostructure of the 3D-dendritic MSNSs with mesopore channels of short length and large diameter?is supposed to be the key role in?immobilization of active and robust heterogeneous catalysts, and?it would have more hopeful prospects in?catalytic applications. (ACS Appl. Mater. Interfaces,?2015,?7, 17450–17459)g.?Visible-light photoredox catalysis?offers exciting opportunities to?achieve challenging carbon–carbon bond formations under mild and ecologically benign conditions. (Acc. Chem. Res.,?2016, 49, 1990–1996)3. 因此同义词：Therefore, thus, consequently, hence, accordingly, so, as a result这一条比较简单，这里主要讲一下这些词的副词词性和灵活运用。

量子力学英语词汇

.量子力学专业英语词汇1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator 线性谐振子41、zero proint energy 零点能.42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢.86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-particle system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles 全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性.。

基于几何光学近似迭代的多重散射波面分析

2021 年 3 月第 44 卷第 2 期湖南师范大学自然科学学报Journal of Natural Science of Hunan Normal University Vel.54 No.2Mar., 2021DOI ： 10.7612/j5ssn.2096W2'l .2021.02.012基于几何光学近似迭代的多重散射波面分析彭梓齐"，杨江河(湖南文理学院数理学院，中国常德415000)摘要为了解析环状光在散射媒质中的传播特性，本文提出了以几何光学近似为基础的波面分析法来进行分析。

该方法主要以几何光学近似法为工具计算散射粒子的前方散射光，并运用迭代计算的方式实现多重散射模型的波面分析。

本文运用该方法计算了散射媒质中的透射率以及环状光在散射媒质中的散射强度波形。

计算结果与实验结果一致,散射媒质在特定的距离以及浓度下,环状光的散射波面中心会出现干涉波峰。

关键词几何光学近似;多重散射;前方散射;干涉中图分类号 O436.1 文献标识码 A 文章编号 2096-5281( 2021) 02-0087-08Wavefrant Analysis of Multiglo Scattering Based onGeometric Optics AppraximationPL#$ Zi-/i ** , B4NG 8a'g-0&收稿日期：2020-07-08基金项目：国家自然科学基金资助项目(U14311⑵；湖南省自然科学基金资助项目(2020JJ5396)；湖南省2018年普通高校教育教学改革研究项目(湘教通[2018]436号)；湖南文理学院博士启动项目(E07018021)* 通信作者，E-mail : pengzq@ (Colleac of Mathematics and Physics , Hunan University of Arts and Science , Changde 415000, China )Abstracr Tv analyzv thv charycte/stics of annular beam propagation in random media , wv proposed a new wavefront analysis method based on geometWc optics approximation. In this algorithm , wv adopted a simplified gev- metic optics approximation and iterative calculation based on forsard scatte/ng and simulated thv attenuation and thv scatw/ng waveform of thv annular beam in random media. An inWr^ered peak was obtained at thv optical axis with a coiain propagation distance and media concentrations , which is consistent with thv expe/mentat results.Key wordt geometWc optics approximation ； multiplv scytte/ng ； forsard scytte/ng ； inte/erenco光学遥感作为成熟的测量技术在工业、医疗、环保等领域得到了广泛应用W 在光学遥感应用中，光在媒质中的传播效率是影响光学测量结果与精度的一项重要参数。

Laser Ranging to the Moon, Mars and Beyond

a r X i v :g r -q c /0411082v 1 16 N o v 2004Laser Ranging to the Moon,Mars and BeyondSlava G.Turyshev,James G.Williams,Michael Shao,John D.AndersonJet Propulsion Laboratory,California Institute of Technology,4800Oak Grove Drive,Pasadena,CA 91109,USAKenneth L.Nordtvedt,Jr.Northwest Analysis,118Sourdough Ridge Road,Bozeman,MT 59715USA Thomas W.Murphy,Jr.Physics Department,University of California,San Diego 9500Gilman Dr.,La Jolla,CA 92093USA Abstract Current and future optical technologies will aid exploration of the Moon and Mars while advancing fundamental physics research in the solar system.Technologies and possible improvements in the laser-enabled tests of various physical phenomena are considered along with a space architecture that could be the cornerstone for robotic and human exploration of the solar system.In particular,accurate ranging to the Moon and Mars would not only lead to construction of a new space communication infrastructure enabling an improved navigational accuracy,but will also provide a signiﬁcant improvement in several tests of gravitational theory:the equivalence principle,geodetic precession,PPN parameters βand γ,and possible variation of the gravitational constant G .Other tests would become possible with an optical architecture that would allow proceeding from meter to centimeter to millimeter range accuracies on interplanetary distances.This paper discusses the current state and the future improvements in the tests of relativistic gravity with Lunar Laser Ranging (LLR).We also consider precision gravitational tests with the future laser rangingto Mars and discuss optical design of the proposed Laser Astrometric Test of Relativity (LATOR)mission.We emphasize that already existing capabilities can oﬀer signiﬁcant improvements not only in the tests of fundamental physics,but may also establish the infrastructure for space exploration in the near future.Looking to future exploration,what characteristics are desired for the next generation of ranging devices,what is the optimal architecture that would beneﬁt both space exploration and fundamental physics,and what fundamental questions can be investigated?We try to answer these questions.1IntroductionThe recent progress in fundamental physics research was enabled by signiﬁcant advancements in many technological areas with one of the examples being the continuing development of the NASA Deep Space Network –critical infrastructure for precision navigation and communication in space.A demonstration of such a progress is the recent Cassini solar conjunction experiment[8,6]that was possible only because of the use of Ka-band(∼33.4GHz)spacecraft radio-tracking capabilities.The experiment was part of the ancillary science program–a by-product of this new radio-tracking technology.Becasue of a much higher data rate transmission and, thus,larger data volume delivered from large distances the higher communication frequency was a very important mission capability.The higher frequencies are also less aﬀected by the dispersion in the solar plasma,thus allowing a more extensive coverage,when depp space navigation is concerned.There is still a possibility of moving to even higher radio-frequencies, say to∼60GHz,however,this would put us closer to the limit that the Earth’s atmosphere imposes on signal transmission.Beyond these frequencies radio communication with distant spacecraft will be ineﬃcient.The next step is switching to optical communication.Lasers—with their spatial coherence,narrow spectral emission,high power,and well-deﬁned spatial modes—are highly useful for many space applications.While in free-space,optical laser communication(lasercomm)would have an advantage as opposed to the conventional radio-communication sercomm would provide not only signiﬁcantly higher data rates(on the order of a few Gbps),it would also allow a more precise navigation and attitude control.The latter is of great importance for manned missions in accord the“Moon,Mars and Beyond”Space Exploration Initiative.In fact,precision navigation,attitude control,landing,resource location, 3-dimensional imaging,surface scanning,formationﬂying and many other areas are thought only in terms of laser-enabled technologies.Here we investigate how a near-future free-space optical communication architecture might beneﬁt progress in gravitational and fundamental physics experiments performed in the solar system.This paper focuses on current and future optical technologies and methods that will advance fundamental physics research in the context of solar system exploration.There are many activities that focused on the design on an optical transceiver system which will work at the distance comparable to that between the Earth and Mars,and test it on the Moon.This paper summarizes required capabilities for such a system.In particular,we discuss how accurate laser ranging to the neighboring celestial bodies,the Moon and Mars,would not only lead to construction of a new space communication infrastructure with much improved navigational accuracy,it will also provide a signiﬁcant improvement in several tests of gravitational theory. Looking to future exploration,we address the characteristics that are desired for the next generation of ranging devices;we will focus on optimal architecture that would beneﬁt both space exploration and fundamental physics,and discuss the questions of critical importance that can be investigated.This paper is organized as follows:Section2discusses the current state and future per-formance expected with the LLR technology.Section3addresses the possibility of improving tests of gravitational theories with laser ranging to Mars.Section4addresses the next logical step—interplanetary laser ranging.We discuss the mission proposal for the Laser Astrometric Test of Relativity(LATOR).We present a design for its optical receiver system.Section5 addresses a proposal for new multi-purpose space architecture based on optical communica-tion.We present a preliminary design and discuss implications of this new proposal for tests of fundamental physics.We close with a summary and recommendations.2LLR Contribution to Fundamental PhysicsDuring more than35years of its existence lunar laser ranging has become a critical technique available for precision tests of gravitational theory.The20th century progress in three seem-ingly unrelated areas of human exploration–quantum optics,astronomy,and human spaceexploration,led to the construction of this unique interplanetary instrument to conduct very precise tests of fundamental physics.In this section we will discuss the current state in LLR tests of relativistic gravity and explore what could be possible in the near future.2.1Motivation for Precision Tests of GravityThe nature of gravity is fundamental to our understanding of the structure and evolution of the universe.This importance motivates various precision tests of gravity both in laboratories and in space.Most of the experimental underpinning for theoretical gravitation has come from experiments conducted in the solar system.Einstein’s general theory of relativity(GR)began its empirical success in1915by explaining the anomalous perihelion precession of Mercury’s orbit,using no adjustable theoretical parameters.Eddington’s observations of the gravitational deﬂection of light during a solar eclipse in1919conﬁrmed the doubling of the deﬂection angles predicted by GR as compared to Newtonian and Equivalence Principle(EP)arguments.Follow-ing these beginnings,the general theory of relativity has been veriﬁed at ever-higher accuracy. Thus,microwave ranging to the Viking landers on Mars yielded an accuracy of∼0.2%from the gravitational time-delay tests of GR[48,44,49,50].Recent spacecraft and planetary mi-crowave radar observations reached an accuracy of∼0.15%[4,5].The astrometric observations of the deﬂection of quasar positions with respect to the Sun performed with Very-Long Base-line Interferometry(VLBI)improved the accuracy of the tests of gravity to∼0.045%[45,51]. Lunar Laser Ranging(LLR),the continuing legacy of the Apollo program,has provided ver-iﬁcation of GR improving an accuracy to∼0.011%via precision measurements of the lunar orbit[62,63,30,31,32,35,24,36,4,68].The recent time-delay experiments with the Cassini spacecraft at a solar conjunction have tested gravity to a remarkable accuracy of0.0023%[8] in measuring deﬂection of microwaves by solar gravity.Thus,almost ninety years after general relativity was born,Einstein’s theory has survived every test.This rare longevity and the absence of any adjustable parameters,does not mean that this theory is absolutely correct,but it serves to motivate more sensitive tests searching for its expected violation.The solar conjunction experiments with the Cassini spacecraft have dramatically improved the accuracy in the solar system tests of GR[8].The reported accuracy of2.3×10−5in measuring the Eddington parameterγ,opens a new realm for gravitational tests,especially those motivated by the on-going progress in scalar-tensor theories of gravity.1 In particular,scalar-tensor extensions of gravity that are consistent with present cosmological models[15,16,17,18,19,20,39]predict deviations of this parameter from its GR value of unity at levels of10−5to10−7.Furthermore,the continuing inability to unify gravity with the other forces indicates that GR should be violated at some level.The Cassini result together with these theoretical predictions motivate new searches for possible GR violations;they also provide a robust theoretical paradigm and constructive guidance for experiments that would push beyond the present experimental accuracy for parameterized post-Newtonian(PPN)parameters(for details on the PPN formalism see[60]).Thus,in addition to experiments that probe the GR prediction for the curvature of the gravityﬁeld(given by parameterγ),any experiment pushingthe accuracy in measuring the degree of non-linearity of gravity superposition(given by anotherEddington parameterβ)will also be of great interest.This is a powerful motive for tests ofgravitational physics phenomena at improved accuracies.Analyses of laser ranges to the Moon have provided increasingly stringent limits on anyviolation of the Equivalence Principle(EP);they also enabled very accurate measurements fora number of relativistic gravity parameters.2.2LLR History and Scientiﬁc BackgroundLLR has a distinguished history[24,9]dating back to the placement of a retroreﬂector array onthe lunar surface by the Apollo11astronauts.Additional reﬂectors were left by the Apollo14and Apollo15astronauts,and two French-built reﬂector arrays were placed on the Moon by theSoviet Luna17and Luna21missions.Figure1shows the weighted RMS residual for each year.Early accuracies using the McDonald Observatory’s2.7m telescope hovered around25cm. Equipment improvements decreased the ranging uncertainty to∼15cm later in the1970s.In1985the2.7m ranging system was replaced with the McDonald Laser Ranging System(MLRS).In the1980s ranges were also received from Haleakala Observatory on the island of Maui in theHawaiian chain and the Observatoire de la Cote d’Azur(OCA)in France.Haleakala ceasedoperations in1990.A sequence of technical improvements decreased the range uncertainty tothe current∼2cm.The2.7m telescope had a greater light gathering capability than thenewer smaller aperture systems,but the newer systemsﬁred more frequently and had a muchimproved range accuracy.The new systems do not distinguish returning photons against thebright background near full Moon,which the2.7m telescope could do,though there are somemodern eclipse observations.The lasers currently used in the ranging operate at10Hz,with a pulse width of about200 psec;each pulse contains∼1018photons.Under favorable observing conditions a single reﬂectedphoton is detected once every few seconds.For data processing,the ranges represented by thereturned photons are statistically combined into normal points,each normal point comprisingup to∼100photons.There are15553normal points are collected until March2004.Themeasured round-trip travel times∆t are two way,but in this paper equivalent ranges in lengthunits are c∆t/2.The conversion between time and length(for distance,residuals,and dataaccuracy)uses1nsec=15cm.The ranges of the early1970s had accuracies of approximately25cm.By1976the accuracies of the ranges had improved to about15cm.Accuracies improvedfurther in the mid-1980s;by1987they were4cm,and the present accuracies are∼2cm.One immediate result of lunar ranging was the great improvement in the accuracy of the lunarephemeris[62]and lunar science[67].LLR measures the range from an observatory on the Earth to a retroreﬂector on the Moon. For the Earth and Moon orbiting the Sun,the scale of relativistic eﬀects is set by the ratio(GM/rc2)≃v2/c2∼10−8.The center-to-center distance of the Moon from the Earth,with mean value385,000km,is variable due to such things as eccentricity,the attraction of the Sun,planets,and the Earth’s bulge,and relativistic corrections.In addition to the lunar orbit,therange from an observatory on the Earth to a retroreﬂector on the Moon depends on the positionin space of the ranging observatory and the targeted lunar retroreﬂector.Thus,orientation ofthe rotation axes and the rotation angles of both bodies are important with tidal distortions,plate motion,and relativistic transformations also coming into play.To extract the gravitationalphysics information of interest it is necessary to accurately model a variety of eﬀects[68].For a general review of LLR see[24].A comprehensive paper on tests of gravitationalphysics is[62].A recent test of the EP is in[4]and other GR tests are in[64].An overviewFigure1:Historical accuracy of LLR data from1970to2004.of the LLR gravitational physics tests is given by Nordtvedt[37].Reviews of various tests of relativity,including the contribution by LLR,are given in[58,60].Our recent paper describes the model improvements needed to achieve mm-level accuracy for LLR[66].The most recent LLR results are given in[68].2.3Tests of Relativistic Gravity with LLRLLR oﬀers very accurate laser ranging(weighted rms currently∼2cm or∼5×10−11in frac-tional accuracy)to retroreﬂectors on the Moon.Analysis of these very precise data contributes to many areas of fundamental and gravitational physics.Thus,these high-precision studies of the Earth-Moon-Sun system provide the most sensitive tests of several key properties of weak-ﬁeld gravity,including Einstein’s Strong Equivalence Principle(SEP)on which general relativity rests(in fact,LLR is the only current test of the SEP).LLR data yielded the strongest limits to date on variability of the gravitational constant(the way gravity is aﬀected by the expansion of the universe),and the best measurement of the de Sitter precession rate.In this Section we discuss these tests in more details.2.3.1Tests of the Equivalence PrincipleThe Equivalence Principle,the exact correspondence of gravitational and inertial masses,is a central assumption of general relativity and a unique feature of gravitation.EP tests can therefore be viewed in two contexts:tests of the foundations of general relativity,or as searches for new physics.As emphasized by Damour[12,13],almost all extensions to the standard modelof particle physics(with best known extension oﬀered by string theory)generically predict newforces that would show up as apparent violations of the EP.The weak form the EP(the WEP)states that the gravitational properties of strong and electro-weak interactions obey the EP.In this case the relevant test-body diﬀerences are their fractional nuclear-binding diﬀerences,their neutron-to-proton ratios,their atomic charges,etc. General relativity,as well as other metric theories of gravity,predict that the WEP is exact. However,extensions of the Standard Model of Particle Physics that contain new macroscopic-range quantumﬁelds predict quantum exchange forces that will generically violate the WEP because they couple to generalized‘charges’rather than to mass/energy as does gravity[17,18]. WEP tests can be conducted with laboratory or astronomical bodies,because the relevant diﬀerences are in the test-body compositions.Easily the most precise tests of the EP are made by simply comparing the free fall accelerations,a1and a2,of diﬀerent test bodies.For the case when the self-gravity of the test bodies is negligible and for a uniform external gravityﬁeld, with the bodies at the same distance from the source of the gravity,the expression for the Equivalence Principle takes the most elegant form:∆a= M G M I 2(1)(a1+a2)where M G and M I represent gravitational and inertial masses of each body.The sensitivity of the EP test is determined by the precision of the diﬀerential acceleration measurement divided by the degree to which the test bodies diﬀer(position).The strong form of the EP(the SEP)extends the principle to cover the gravitational properties of gravitational energy itself.In other words it is an assumption about the way that gravity begets gravity,i.e.about the non-linear property of gravitation.Although general relativity assumes that the SEP is exact,alternate metric theories of gravity such as those involving scalarﬁelds,and other extensions of gravity theory,typically violate the SEP[30,31, 32,35].For the SEP case,the relevant test body diﬀerences are the fractional contributions to their masses by gravitational self-energy.Because of the extreme weakness of gravity,SEP test bodies that diﬀer signiﬁcantly must have astronomical sizes.Currently the Earth-Moon-Sun system provides the best arena for testing the SEP.The development of the parameterized post-Newtonian formalism[31,56,57],allows one to describe within the common framework the motion of celestial bodies in external gravitational ﬁelds within a wide class of metric theories of gravity.Over the last35years,the PPN formalism has become a useful framework for testing the SEP for extended bodies.In that formalism,the ratio of passive gravitational to inertial mass to theﬁrst order is given by[30,31]:M GMc2 ,(2) whereηis the SEP violation parameter(discussed below),M is the mass of a body and E is its gravitational binding or self-energy:E2Mc2 V B d3x d3yρB(x)ρB(y)EMc2 E=−4.64×10−10andwhere the subscripts E and m denote the Earth and Moon,respectively.The relatively small size bodies used in the laboratory experiments possess a negligible amount of gravitational self-energy and therefore such experiments indicate nothing about the equality of gravitational self-energy contributions to the inertial and passive gravitational masses of the bodies [30].TotesttheSEP onemustutilize planet-sizedextendedbodiesinwhichcase theratioEq.(3)is considerably higher.Dynamics of the three-body Sun-Earth-Moon system in the solar system barycentric inertial frame was used to search for the eﬀect of a possible violation of the Equivalence Principle.In this frame,the quasi-Newtonian acceleration of the Moon (m )with respect to the Earth (E ),a =a m −a E ,is calculated to be:a =−µ∗rM I m µS r SEr 3Sm + M G M I m µS r SEr 3+µS r SEr 3Sm +η E Mc 2 m µS r SEMc 2 E − E n 2−(n −n ′)2n ′2a ′cos[(n −n ′)t +D 0].(8)Here,n denotes the sidereal mean motion of the Moon around the Earth,n ′the sidereal mean motion of the Earth around the Sun,and a ′denotes the radius of the orbit of the Earth around the Sun (assumed circular).The argument D =(n −n ′)t +D 0with near synodic period is the mean longitude of the Moon minus the mean longitude of the Sun and is zero at new Moon.(For a more precise derivation of the lunar range perturbation due to the SEP violation acceleration term in Eq.(6)consult [62].)Any anomalous radial perturbation will be proportional to cos D .Expressed in terms ofη,the radial perturbation in Eq.(8)isδr∼13ηcos D meters [38,21,22].This eﬀect,generalized to all similar three body situations,the“SEP-polarization eﬀect.”LLR investigates the SEP by looking for a displacement of the lunar orbit along the direction to the Sun.The equivalence principle can be split into two parts:the weak equivalence principle tests the sensitivity to composition and the strong equivalence principle checks the dependence on mass.There are laboratory investigations of the weak equivalence principle(at University of Washington)which are about as accurate as LLR[7,1].LLR is the dominant test of the strong equivalence principle.The most accurate test of the SEP violation eﬀect is presently provided by LLR[61,48,23],and also in[24,62,63,4].Recent analysis of LLR data test the EP of∆(M G/M I)EP=(−1.0±1.4)×10−13[68].This result corresponds to a test of the SEP of∆(M G/M I)SEP=(−2.0±2.0)×10−13with the SEP violation parameter η=4β−γ−3found to beη=(4.4±4.5)×10−ing the recent Cassini result for the PPN parameterγ,PPN parameterβis determined at the level ofβ−1=(1.2±1.1)×10−4.2.3.2Other Tests of Gravity with LLRLLR data yielded the strongest limits to date on variability of the gravitational constant(the way gravity is aﬀected by the expansion of the universe),the best measurement of the de Sitter precession rate,and is relied upon to generate accurate astronomical ephemerides.The possibility of a time variation of the gravitational constant,G,wasﬁrst considered by Dirac in1938on the basis of his large number hypothesis,and later developed by Brans and Dicke in their theory of gravitation(for more details consult[59,60]).Variation might be related to the expansion of the Universe,in which case˙G/G=σH0,where H0is the Hubble constant, andσis a dimensionless parameter whose value depends on both the gravitational constant and the cosmological model considered.Revival of interest in Brans-Dicke-like theories,with a variable G,was partially motivated by the appearance of superstring theories where G is considered to be a dynamical quantity[26].Two limits on a change of G come from LLR and planetary ranging.This is the second most important gravitational physics result that LLR provides.GR does not predict a changing G,but some other theories do,thus testing for this eﬀect is important.The current LLR ˙G/G=(4±9)×10−13yr−1is the most accurate limit published[68].The˙G/G uncertaintyis83times smaller than the inverse age of the universe,t0=13.4Gyr with the value for Hubble constant H0=72km/sec/Mpc from the WMAP data[52].The uncertainty for˙G/G is improving rapidly because its sensitivity depends on the square of the data span.This fact puts LLR,with its more then35years of history,in a clear advantage as opposed to other experiments.LLR has also provided the only accurate determination of the geodetic precession.Ref.[68]reports a test of geodetic precession,which expressed as a relative deviation from GR,is K gp=−0.0019±0.0064.The GP-B satellite should provide improved accuracy over this value, if that mission is successfully completed.LLR also has the capability of determining PPNβandγdirectly from the point-mass orbit perturbations.A future possibility is detection of the solar J2from LLR data combined with the planetary ranging data.Also possible are dark matter tests,looking for any departure from the inverse square law of gravity,and checking for a variation of the speed of light.The accurate LLR data has been able to quickly eliminate several suggested alterations of physical laws.The precisely measured lunar motion is a reality that any proposed laws of attraction and motion must satisfy.The above investigations are important to gravitational physics.The future LLR data will improve the above investigations.Thus,future LLR data of current accuracy would con-tinue to shrink the uncertainty of˙G because of the quadratic dependence on data span.The equivalence principle results would improve more slowly.To make a big improvement in the equivalence principle uncertainty requires improved range accuracy,and that is the motivation for constructing the APOLLO ranging facility in New Mexico.2.4Future LLR Data and APOLLO facilityIt is essential that acquisition of the new LLR data will continue in the future.Accuracies∼2cm are now achieved,and further very useful improvement is expected.Inclusion of improved data into LLR analyses would allow a correspondingly more precise determination of the gravitational physics parameters under study.LLR has remained a viable experiment with fresh results over35years because the data accuracies have improved by an order of magnitude(see Figure1).There are prospects for future LLR station that would provide another order of magnitude improvement.The Apache Point Observatory Lunar Laser-ranging Operation(APOLLO)is a new LLR eﬀort designed to achieve mm range precision and corresponding order-of-magnitude gains in measurements of fundamental physics parameters.For theﬁrst time in the LLR history,using a3.5m telescope the APOLLO facility will push LLR into a new regime of multiple photon returns with each pulse,enabling millimeter range precision to be achieved[29,66].The anticipated mm-level range accuracy,expected from APOLLO,has a potential to test the EP with a sensitivity approaching10−14.This accuracy would yield sensitivity for parameterβat the level of∼5×10−5and measurements of the relative change in the gravitational constant,˙G/G, would be∼0.1%the inverse age of the universe.The overwhelming advantage APOLLO has over current LLR operations is a3.5m astro-nomical quality telescope at a good site.The site in southern New Mexico oﬀers high altitude (2780m)and very good atmospheric“seeing”and image quality,with a median image resolu-tion of1.1arcseconds.Both the image sharpness and large aperture conspire to deliver more photons onto the lunar retroreﬂector and receive more of the photons returning from the re-ﬂectors,pared to current operations that receive,on average,fewer than0.01 photons per pulse,APOLLO should be well into the multi-photon regime,with perhaps5–10 return photons per pulse.With this signal rate,APOLLO will be eﬃcient atﬁnding and track-ing the lunar return,yielding hundreds of times more photons in an observation than current√operations deliver.In addition to the signiﬁcant reduction in statistical error(useful).These new reﬂectors on the Moon(and later on Mars)can oﬀer signiﬁcant navigational accuracy for many space vehicles on their approach to the lunar surface or during theirﬂight around the Moon,but they also will contribute signiﬁcantly to fundamental physics research.The future of lunar ranging might take two forms,namely passive retroreﬂectors and active transponders.The advantages of new installations of passive retroreﬂector arrays are their long life and simplicity.The disadvantages are the weak returned signal and the spread of the reﬂected pulse arising from lunar librations(apparent changes in orientation of up to10 degrees).Insofar as the photon timing error budget is dominated by the libration-induced pulse spread—as is the case in modern lunar ranging—the laser and timing system parameters do√not inﬂuence the net measurement uncertainty,which simply scales as1/3Laser Ranging to MarsThere are three diﬀerent experiments that can be done with accurate ranges to Mars:a test of the SEP(similar to LLR),a solar conjunction experiment measuring the deﬂection of light in the solar gravity,similar to the Cassini experiment,and a search for temporal variation in the gravitational constant G.The Earth-Mars-Sun-Jupiter system allows for a sensitive test of the SEP which is qualitatively diﬀerent from that provided by LLR[3].Furthermore,the outcome of these ranging experiments has the potential to improve the values of the two relativistic parameters—a combination of PPN parametersη(via test of SEP)and a direct observation of the PPN parameterγ(via Shapiro time delay or solar conjunction experiments).(This is quite diﬀerent compared to LLR,as the small variation of Shapiro time delay prohibits very accurate independent determination of the parameterγ).The Earth-Mars range would also provide for a very accurate test of˙G/G.This section qualitatively addresses the near-term possibility of laser ranging to Mars and addresses the above three eﬀects.3.1Planetary Test of the SEP with Ranging to MarsEarth-Mars ranging data can provide a useful estimate of the SEP parameterηgiven by Eq.(7). It was demonstrated in[3]that if future Mars missions provide ranging measurements with an accuracy ofσcentimeters,after ten years of ranging the expected accuracy for the SEP parameterηmay be of orderσ×10−6.These ranging measurements will also provide the most accurate determination of the mass of Jupiter,independent of the SEP eﬀect test.It has been observed previously that a measurement of the Sun’s gravitational to inertial mass ratio can be performed using the Sun-Jupiter-Mars or Sun-Jupiter-Earth system[33,47,3]. The question we would like to answer here is how accurately can we do the SEP test given the accurate ranging to Mars?We emphasize that the Sun-Mars-Earth-Jupiter system,though governed basically by the same equations of motion as Sun-Earth-Moon system,is signiﬁcantly diﬀerent physically.For a given value of SEP parameterηthe polarization eﬀects on the Earth and Mars orbits are almost two orders of magnitude larger than on the lunar orbit.Below we examine the SEP eﬀect on the Earth-Mars range,which has been measured as part of the Mariner9and Viking missions with ranging accuracy∼7m[48,44,41,43].The main motivation for our analysis is the near-future Mars missions that should yield ranging data, accurate to∼1cm.This accuracy would bring additional capabilities for the precision tests of fundamental and gravitational physics.3.1.1Analytical Background for a Planetary SEP TestThe dynamics of the four-body Sun-Mars-Earth-Jupiter system in the Solar system barycentric inertial frame were considered.The quasi-Newtonian acceleration of the Earth(E)with respect to the Sun(S),a SE=a E−a S,is straightforwardly calculated to be:a SE=−µ∗SE·r SE MI Eb=M,Jµb r bS r3bE + M G M I E b=M,Jµb r bS。

基于自转一阶非连续式微球双平盘研磨的运动学分析与实验研究

第53卷第8期表面技术2024年4月SURFACE TECHNOLOGY·133·基于自转一阶非连续式微球双平盘研磨的运动学分析与实验研究吕迅1,2*，李媛媛1，欧阳洋1，焦荣辉1，王君1，杨雨泽1（1.浙江工业大学机械工程学院，杭州 310023；2.新昌浙江工业大学科学技术研究院，浙江绍兴 312500）摘要：目的分析不同研磨压力、下研磨盘转速、保持架偏心距和固着磨料粒度对微球精度的影响，确定自转一阶非连续式双平面研磨方式在加工GCr15轴承钢球时的最优研磨参数，提高微球的形状精度和表面质量。

方法首先对自转一阶非连续式双平盘研磨方式微球进行运动学分析，引入滑动比衡量微球在不同摩擦因数区域的运动状态，建立自转一阶非连续式双平盘研磨方式下的微球轨迹仿真模型，利用MATLAB对研磨轨迹进行仿真，分析滑动比对研磨轨迹包络情况的影响。

搭建自转一阶非连续式微球双平面研磨方式的实验平台，采用单因素实验分析主要研磨参数对微球精度的影响，得到考虑圆度和表面粗糙度的最优参数组合。

结果实验结果表明，在研磨压力为0.10 N、下研磨盘转速为20 r/min、保持架偏心距为90 mm、固着磨料粒度为3000目时，微球圆度由研磨前的1.14 μm下降至0.25 μm，表面粗糙度由0.129 1 μm下降至0.029 0 μm。

结论在自转一阶非连续式微球双平盘研磨方式下，微球自转轴方位角发生突变，使研磨轨迹全覆盖在球坯表面。

随着研磨压力、下研磨盘转速、保持架偏心距的增大，微球圆度和表面粗糙度呈现先降低后升高的趋势。

随着研磨压力与下研磨盘转速的增大，材料去除速率不断增大，随着保持架偏心距的增大，材料去除速率降低。

随着固着磨料粒度的减小，微球的圆度和表面粗糙度降低，材料去除速率降低。

关键词：自转一阶非连续；双平盘研磨；微球；运动学分析；研磨轨迹；研磨参数中图分类号：TG356.28 文献标志码：A 文章编号：1001-3660(2024)08-0133-12DOI：10.16490/ki.issn.1001-3660.2024.08.012Kinematic Analysis and Experimental Study of Microsphere Double-plane Lapping Based on Rotation Function First-order DiscontinuityLYU Xun1,2*, LI Yuanyuan1, OU Yangyang1, JIAO Ronghui1, WANG Jun1, YANG Yuze1(1. College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310023, China;2. Xinchang Research Institute of Zhejiang University of Technology, Zhejiang Shaoxing 312500, China)ABSTRACT: Microspheres are critical components of precision machinery such as miniature bearings and lead screws. Their surface quality, roundness, and batch consistency have a crucial impact on the quality and lifespan of mechanical parts. Due to收稿日期：2023-07-28；修订日期：2023-09-26Received：2023-07-28；Revised：2023-09-26基金项目：国家自然科学基金（51975531）Fund：National Natural Science Foundation of China (51975531)引文格式：吕迅, 李媛媛, 欧阳洋, 等. 基于自转一阶非连续式微球双平盘研磨的运动学分析与实验研究[J]. 表面技术, 2024, 53(8): 133-144.LYU Xun, LI Yuanyuan, OU Yangyang, et al. Kinematic Analysis and Experimental Study of Microsphere Double-plane Lapping Based on Rotation Function First-order Discontinuity[J]. Surface Technology, 2024, 53(8): 133-144.*通信作者（Corresponding author）·134·表面技术 2024年4月their small size and light weight, existing ball processing methods are used to achieve high-precision machining of microspheres. Traditional concentric spherical lapping methods, with three sets of circular ring trajectories, result in poor lapping accuracy. To achieve efficient and high-precision processing of microspheres, the work aims to propose a method based on the first-order discontinuity of rotation for double-plane lapping of microspheres. Firstly, the principle of the first-order discontinuity of rotation for double-plane lapping of microspheres was analyzed, and it was found that the movement of the microsphere changed when it was in different regions of the upper variable friction plate, resulting in a sudden change in the microsphere's rotational axis azimuth and expanding the lapping trajectory. Next, the movement of the microsphere in the first-order discontinuity of rotation for double-plane lapping method was analyzed, and the sliding ratio was introduced to measure the motion state of the microsphere in different friction coefficient regions. It was observed that the sliding ratio of the microsphere varied in different friction coefficient regions. As a result, when the microsphere passed through the transition area between the large and small friction regions of the upper variable friction plate, the sliding ratio changed, causing a sudden change in the microsphere's rotational axis azimuth and expanding the lapping trajectory. The lapping trajectory under different sliding ratios was simulated by MATLAB, and the results showed that with the increase in simulation time, the first-order discontinuity of rotation for double-plane lapping method could achieve full coverage of the microsphere's lapping trajectory, making it more suitable for precision machining of microspheres. Finally, based on the above research, an experimental platform for the first-order discontinuity of rotation for double-plane lapping of microsphere was constructed. With 1 mm diameter bearing steel balls as the processing object, single-factor experiments were conducted to study the effects of lapping pressure, lower plate speed, eccentricity of the holding frame, and grit size of fixed abrasives on microsphere roundness, surface roughness, and material removal rate. The experimental results showed that under the first-order discontinuity of rotation for double-plane lapping, the microsphere's rotational axis azimuth underwent a sudden change, leading to full coverage of the lapping trajectory on the microsphere's surface. Under the lapping pressure of 0.10 N, the lower plate speed of 20 r/min, the eccentricity of the holder of 90 mm, and the grit size of fixed abrasives of 3000 meshes, the roundness of the microsphere decreased from 1.14 μm before lapping to 0.25 μm, and the surface roughness decreased from 0.129 1 μm to 0.029 0 μm. As the lapping pressure and lower plate speed increased, the microsphere roundness and surface roughness were firstly improved and then deteriorated, while the material removal rate continuously increased. As the eccentricity of the holding frame increased, the roundness was firstly improved and then deteriorated, while the material removal rate decreased. As the grit size of fixed abrasives decreased, the microsphere's roundness and surface roughness were improved, and the material removal rate decreased. Through the experiments, the optimal parameter combination considering roundness and surface roughness is obtained: lapping pressure of 0.10 N/ball, lower plate speed of 20 r/min, eccentricity of the holder of 90 mm, and grit size of fixed abrasives of 3000 meshes.KEY WORDS: rotation function first-order discontinuity; double-plane lapping; microsphere; kinematic analysis; lapping trajectory; lapping parameters随着机械产品朝着轻量化、微型化的方向发展，微型电机、仪器仪表等多种工业产品对微型轴承的需求大量增加。

科技英语_秦荻辉_科技英语语法习题以及答案

科技英语_秦荻辉_科技英语语法习题以及答案练习 1I、将下列句子译成汉语，注意句中有些冠词的特殊位置:1. In this case the current(电流)exists for only half the cycle(周期).2. In such a case there is no current flowing in the circuit(电路).3. Sensitivity(灵敏度)is a measure of how small a signal(信号)a receiver(接收机)canpick up and amplify(放大)to a level useful for communications.4. ε may be as small a positive constant as you please.5. Even so fundamental a dimension,量纲，as time was measured extremely crudely with sandand water clocks hundreds of years ago.6. Nonlinear distortion,非线性失真，can be caused by too large an input signal.7. The method used is quite an effective one.8. A series,级数，solution of this kind of problem allows as close a calculation of the error as needed.II、将下列句子译成汉语，注意句中“and”和“or”的确切含义:1. Air has weight and occupies space.2. In this way less collector dissipation(集电极功耗)results, and the efficiency increases.3. We can go one step farther and take into account the nonzeroslope of the actual curves.4. Try hard, and you will work the nut(螺母)loose.5. The first step in analyzing a physical situation is to select those aspects of it which are essential and disregard the others.6. This satellite was used for communications between the United States and Great Britain, France and Italy.7. Some physical quantities require only a magnitude and a unit tobe completely specified. Thus it is sufficient to say that the mass of a man is 85 kg, that the area of a farm is 160 acres, that the frequencyof a sound wave is 660 cycles/sec, and that a light bulb consumes electric energy at the rate of 100 watts.8. Geothermal energy, or energy from within the earth, can be usedto generate electricity.o9.The current in a capacitor(电容器)leads(导前)the voltage by 90, or, the voltage lagsothe current by 90.10. The message is a logical unit of user data, control data, or both.III、将下列句子译成汉语，注意句中分数和倍数的正确译法:1. By varying V only a few hundredths of a volt, the base current(基极电流)can be BEchanged significantly.2. The standard meter is accurate to about two parts in one billion.3. Cromatographic(层析的)techniques have been developed to detectair pollutants atconcentrations(浓度)of one part per million or less.4. The volume coefficient(体膨胀系数)of a solid is almost exactlythree times its linearcoefficient.5. The demand for this kind of equipment in the near future will be20 times what it is.6. The wavelength of this musical note(音符)is7.8 ft, over threetimes longer than thewavelength of the same note in air (2.5 ft).7. This causes the collector current(集电极电流)to change by afactor of approximately β.8. This factor(因子)is now equal to 9, a reduction by a factor of 11.IV、将下列句子译成英语:1、火箭是由金属制成的。

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a r X i v :g r -q c /0212063v 2 28 M a r 2003Generalizations of pp-wave spacetimes in higher dimensionsA.Coley †,son †,N.Pelavas †,V.Pravda ‡,A.Pravdov´a ‡and R.Zalaletdinov ††Department of Mathematics and Statistics,Dalhousie University,Halifax,Nova Scotia ‡Mathematical Institute,Academy of Sciences,ˇZitn´a 25,11567Prague 1,Czech Republic We shall investigate D -dimensional Lorentzian spacetimes in which all of the scalar invariantsconstructed from the Riemann tensor and its covariant derivatives are zero.These spacetimes arehigher-dimensional generalizations of D -dimensional pp-wave spacetimes,which have been of interestrecently in the context of string theory in curved backgrounds in higher dimensions.I.INTRODUCTIONHigher-dimensional pp-wave spacetimes are of current interest in string theory in curved backgrounds,particularly since these Lorentzian spacetimes are exact solutions in string theory and their spectrum can therefore be explicitly determined.In this paper we shall discuss D -dimensional Lorentzian spacetimes in which all of the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero.These spacetimes can be regarded as higher-dimensional generalizations of D -dimensional pp-wave spacetimes.This research follows on from the recent work of [1]in four dimensions,in which it was proven that in Lorentzian spacetimes all of the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero if and only if the spacetime is of Petrov type III,N or O,all eigenvalues of the Ricci tensor are zero and the common multiple null eigenvector l a of the Weyl and Ricci tensors is geodesic,shearfree,non-expanding,and non-twisting [1](i.e.,the Newman-Penrose (NP)coeﬃcients κ,σ,and ρare zero);we shall refer to these spacetimes as vanishing scalar invariant (VSI)spacetimes.The Ricci tensor has the form R ab =−2Al a l b +4A i l (a m i b )(1)(i =1,2).The Pleba´n ski-Petrov type (PP-type)is N for A i =0or O for A i =0.We note that for PP-type N,using a null rotation a boost and a spatial rotation we can transform away the Ricci component A and set A i =1.For PP-type O it is possible to set A =1by performing a boost.It is known that the energy-momentum tensor for a spacetime corresponding to PP-type N cannot satisfy the weak energy conditions [2],and hence such spacetimes of are not regarded as physical in classical general relativity (however,see [3]).Therefore,attention is usually restricted to PP-type O models,in which the energy-momentum tensor corresponds to a pure null radiation ﬁeld [2].All of these spacetimes belong to Kundt’s class,and hence the metric of these spacetimes can be expressed in an appropriate form in adapted coordinates [2,4].The metrics for all VSI spacetimes are displayed in [1].The generalized pp-wave solutions are of Petrov-type N,PP-type O (so that the Ricci tensor has the form of null radiation)with τ=0,and admit a covariantly constant null vector ﬁeld [5].The vacuum spacetimes,which are obtained by setting A =0,are the well-known pp-wave spacetimes (or plane-frontedgravitational waves with parallel rays).II.HIGHER ORDER THEOREMThis theorem can be readily generalized to higher dimensions.We shall study Lorentzian VSI spacetimes in arbitrary D -dimensions (not necessary even,but D =10is of particular importance from string theory)with signature D −2.In principle we could study other signatures;for example,manifolds with signature D −4with D ≥5may also be of physical interest [6].Let the tetrad be l,n,m 1,m 2,...,m i (l,n null with l a l a =n a n a =0,l a n a =1,m i real and spacelike),so thatg ab =2l (a n b )+δjk m j a m k b .(2)Using the notation{[w p x q ][y r z s ]}≡w p x q y r z s −w p x q z r y s −x p w q y r z s +x p w q z r y s +y p z q w r x s −y p z q x r w s −z p y q w r x s +z p y q x r w s ,(3)if all zeroth order invariants vanish then there exists a null tetrad (2)l,n,m i (i =1,...,N =D −2)such that [7]1R abcd=A i{[l a n b][l c m i d]}+B[ij]k{[m i a m jb ][lc m k d]}+C(ij){[l a m i b][l c m jd]}.(4)We still have the freedom to“choose the frame”and simplify further,using boosts,spins and null rotations,depending on the algebraic structure of the Ricci and Weyl tensors[a generalization of Petrov and(Petrov-Plebanski)PP classiﬁcations].From(4)we obtain the Ricci tensor:R bd=[−A i+2B[ij]kδjk](l b m i d+l d m i b)+Al b l d,(5) where A≡2C jkδjk.We can further simplify R bd depending on its algebraic type.If the energy conditions are satisﬁedA i−2B[ij]kδjk=0,(6) we shall refer to this as type P PN.From(4)we obtain the Weyl tensor:C abcd=Ψi{[l a n b][l c m i d]}+Ψ{ijk}{[m i a m jb ][lc m k d]}+Ψ<i j>{[l a m i b][l c m jd]},(8)whereΨi=2Ψ{ijk}δjk≡C abcd n a l b n c m d i=12C abcd m a i m b j n c m d k=B[ij]k+12C abcd n a m b i n c m j d=C(i j)−1III(in someinstances we can use the remaining tetrad freedom in this case to setΨ<i j>=0).Spacetimes withΨ{ijk}=0will be referred to as of type PO.[Ψ{ijk}andΨ<i j>are higher-dimensional analogues of the complex NP coeﬃcents Ψ4in4dimensions.A comprehensive higher-dimensional Petrov classiﬁcation,which is not necessary here, will be discussed elsewhere.]For spacetimes of type P P N,the Ricci tensor is given by(7)and the Weyl tensor is given byC abcd=[C(ij)−1d s2=−2d u[H d u+d v+W i d x i]+g ij d x i d x j,(10) where i,j,k=1,...,N and the metric functionsH=H(u,v,x i),W i=W i(u,v,x i),g ij=g ij(u,x i)satisfy the remaining vanishing invariant conditions and the Einsteinﬁeld equations(see[2,4]).We may,without loss of generality,use the remaining coordinate freedom(e.g.,transformations of the form x i′=x i′(u,x j))to simplify g ij.For the spacetimes considered here we shall diagonalize g ij,and in the particular examples below we shall take g ij=δij.The null tetrad is thenl=−∂v,n=∂u−(H+1O and type Px1,W i=0(i=1),H=−ǫv2III(“τ=0”)solutionsg ij=δij,W i=ǫW i(u,x k),H=ǫvH1(u,x k)+H0(u,x k).(13) In general these spacetimes are of type P PO(null radiation)the functions W i(u,x k)and H1(u,x k)satisfy additional diﬀerential equations.The higher-dimensional type PRecently the idea that our universe is embedded in a higher-dimensional world has received a great deal of renewed attention[22].Due to the importance of branes in understanding the non-perturbative dynamics of string theories, a number of classical solutions of branes in the background of a pp-wave have been studied;in particular a new brane-world model has been introduced in which the bulk solution consists of outgoing plane waves(only)[23].For example,a class of pp-wave string solutions with non-constant NS-NS or R-Rﬁeld strengths,which are exact type II superstring solutions to all orders inα′since all corrections to the leading-orderﬁeld equations naturally vanish,were discussed recently[14](see also[24]).The metric ansatz and NS-NS2-form potential in10-dimensional superstring theory is given byds2=−dudv−K(x k)du2+dx2i+dy2m,B2=b m(x k)du∧dy m,H3=∂i b m(x k)dx i∧du∧dy m,(14) where i=1,...,d and m=d+1,..,8(and a dilaton of the formφ=ρi x i+˜φ(u)can be included).In particular,it was found[14]that the only non-zero component of the generalized curvature isR uiuj=−12∂i b m∂j b m.(15)These solutions are consequently of type P P N(see(7)and(9)).There are several special cases.For b m=0the standard higher-dimensional generalized pp-wave solution is recovered with K=K0(x)being a harmonic function.WZW models[25]result when the b m are linear,corresponding to homogeneous plane-wave backgrounds with constant H3ﬁeld.The Laplace equation for b m can also be solved by choosing b m to be the real part of complex holomorphic functions.The R-R counterparts of these string models are direct analogs of the pp-wave solution[24] supported by a non-constant5-form background.Note that lifts of the above solutions to11dimensions belong to a class of D=11pp-wave backgroundsﬁrst considered in[26].Acknowledgements.This work was supported,in part,by the Natural Sciences and Engineering Research Council of Canada.[21]J.Kowalski-Glikman,Phys.Lett.B150,125(1985).[22]V.Rubakov and M.Shaposhnikov,Phys.Lett.B125,139(1983);N.Arkani-Hamed,S.Dimopoulos and G.Dvali,Phys.Lett.B429,263(1998);L.Randall and R.Sundrum,Phys.Rev.Lett.833370&4690(1999).[23]G.T.Horowitz,I.Low and A.Zee,Preprint:hep-th/0004206.[24]J.Maldacena and L.Maoz,Preprint:hep-th/0207284.[25]C.R.Nappi and E.Witten,Phys.Rev.Lett.71,3751(1993).[26]C.M.Hull,Phys.Lett.B139,39(1984).5。