Generalizations of pp-wave spacetimes in higher dimensions

合集下载

谱稀疏ppt

Spectral Compressive Sensing
• Compressive sensing applied to frequency-sparse signals
=
frequency-sparse signal
linear measurements
nonzero DFT coefficients
1
µ( )
0.5 0 0 1
0.02
0.04
0.06
0.08
0.1
0.12
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.
Normalized approx. error
1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 Approximation sparsity K 50 Integral frequencies Arbitrary frequencies
Signal is sum of 10 sinusoids
RN
RM
K
K
x1
x2
mK K-dim planes
[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。

肖体乔_戚俊成博士学位论文2022

Shanghai Institute of Applied Physics Chinese Academy of Sciences
Aprial 2014
III
IV
V
同步辐射 X 射线光栅成像及其在相干性测量中的应用研究
戚俊成
导师：肖体乔研究员
摘要
自 X 射线光栅成像出现以来，迅速成为了 X 射线成像领域的研究热点。由于通过一套原始数据就可以获得样品的吸收、相位和暗场信息，既保持了传统 X 射线吸收成像的优点，又结合了暗场成像和相位衬度成像的优势，这三者各自可以独立地展示样品的信息，同时三者之间又互为补充。具有重要的研究价值和广泛的应用前景。
Qi Juncheng
A Dissertation Submitted to The University of Chinese Academy of Sciences
In partial fulfillment of the requirement For the degree of
Doctor of engineering
本研究工作基于上海光源 X 射线成像及其生物医学应用光束线站(BL13W) 的实验平台，建立了上海光源 X 射线光栅成像系统并对其进行了优化，同时基于 X 射线光栅干涉仪对上海光源光束相干特性进行了研究，并取得了以下几方面的成果： 1. 开展了基于上海光源 X 射线成像及其生物医学应用光束线的光栅成像方面
Imaging and Biomedical Application beamline of (SSRF). Firstly, establish the grating based X-ray imaging setup, and code the data process software; And then do same research about quantitative study and medical application. All the results shown that the feasibility of the system and can be open for user. 2. Optimized the Grating based X-ray imaging system based EST algorithm. By the means of translation of the coordinates, transform the rotation variable differential phase data to rotation invariable data, which can apply EST algorithm to reconstruction. The simulation and the experimental result testified the feasibility of the proposed method, and the method can

Towards S matrices on flat space and pp waves from SYM

a r X i v :h e p -t h /0501013v 1 3 J a n 2005hep-th/0501013BROWN-HET-1427Towards S matrices on ﬂat space and pp waves from SYMAntal Jevicki and Horatiu Nastase Brown University Providence,RI,02912,USA Abstract We analyze the possibility of extracting S matrices on pp waves and ﬂat space from SYM correlators.For pp waves,there is a sub-tlety in deﬁning S matrices,but we can certainly obtain observables.Only extremal correlators survive the pp wave limit.A ﬁrst quan-tized string approach is inconclusive,producing in the simplest form results that vanish in the pp wave limit.We deﬁne a procedure to get S matrices from SYM correlators,both for ﬂat space and for pp waves,generalizing a procedure due to Giddings.We analyze nonrenormalized correlators:2and 3-point functions and extremal correlators.For the extremal 3-point function,the SYM and AdS results for the S matrix match for the angular dependence,but the energy dependence doesn’t.1IntroductionOne can understand holography rather straightforwardly in the usual AdS-CFT correspon-dence[1].The boundary of global AdS5×S5space is S3×R t.Supergravity correlators in the bulk,with sources living on the boundary,are the same as SYM correlators on the boundary[2,3].SYM operators on the R4plane correspond by conformal invariance to states on S3×R t,and are mapped by AdS-CFT to normalizable modes in global AdS5×S5. The size of AdS5×S5in string units is given by the’t Hooft coupling,R/√There is one more problem with this fact though∗.Namely,there is a subtlety in the deﬁnition of S matrices for massless external states.On the pp wave,string states have−2p−=µ i N0i+µ i n i (µα′p+)2= i p2i p+(1.1)where we have rewritten theﬁrst term in terms of discrete momenta p i that will become continous in theﬂat space limit of the pp wave(µ→0).To construct the S matrix weﬁrst construct wavepacketsφ= n dp−√2π dp+2p+φ n(p+,p−)| n,p+,p−>(1.3) The boundary of the pp wave space is a one dimensional null line,parametrized by x+ [13],and so asymptotic states can only be deﬁned in one direction(x−),parametrized by p+.Indeed,in the transverse directions(x i),there is a harmonic oscillator potential,thus states are just parametrized by oscillator numbers,but are not asymptotic,and then the energy p−is derived on-shell from them.Thus[25]deﬁne S matrices as scatterings in a two dimensional eﬀective theory(p+,p−),with extra discrete indices(oscillators):L2= nφ∗ n(p+,p−)(p−−E( n,p+))φ n(p+,p−)+L int(1.4)Note though that now the dispersion relation(1.1)at M=0has no p+dependence(in the form after theﬁrst equal sign),i.e.∂p−/∂p+=0so no group velocity,thus one cannot kick the waves[25].Another way to spot the problem at M=0is to try to write down cross sections using the wavepackets.Following what happens inﬂat space when we use wavepackets to turn S matrices to cross-sections,we see the problem:S matrices have overall momentum delta functionsδd( p i),but external particles are on shell,thus integrations are over d d−1k j. Usually,the last integration in dσis of the typedp1p d−21dΩ∗We thank Juan Maldacena for pointing this out to us.See also the Note added in[25]always mimic what happens in theﬂat space limit of the pp wave,as we did in the second√equality in(1.1).That is,write down wavepackets in the discrete momenta p i=2Extremal correlators and the Penrose limitIn this section we explain and expand the points made in[13]about the importance of extremal correlators in the Penrose limit.In order to deﬁne S matrices on the pp wave we need to understand the relation between the normalizable modes on the pp wave and states in SYM.The pp wave metric isds2=2dx+dx−−µ2r2(dx+)2+dx i dx i(2.1) where r2=x i x i.The normalized solutions to the wave equation on the pp wave(2−m2)φ=(2∂+∂−+µ2r2∂2−+∂2i−m2)φ=0(2.2) areφ=φ(x+)ψ(x−)ψT(x i)=(e ip−x+)(Be ip+x−) i((p+)1/4√p+x i))(2.3)where−2p+p−=2µp+8 i=1(n i+1p+delta function(in momentum)normalization.But the pp wave limit comes from the(modiﬁed)Penrose limit of AdS,where p+=J/R2,as x−is a circle of radius R2.Therefore,when we calculate SYM amplitudes rather than gravity amplitudes,we have to use the compact normalization(δJ1J2)for states,thusB C=1p+R=B NCthat will give a3-point function that behaves as(keeping only p+factors and dropping the +index)A NC∼g s p3/2δ(p3−p1−p2)(2.9) It will convert in SYM variables toA C∼J3/2k1k2k3/N,thus if all the k’s are of order J,the correlator is of order J3/2/N as we claimed that we need,and there would be no problem.There is also an invariant tensor<C I1C I2C I3> that would need to be rescaled,but in the general case it would not bring in new powers of J.However,in the Penrose limit we are interested in operators that have mostly Z insertions (where Z=Φ5+iΦ6is a complex scalar)and only a fewΦi“impurities”(i=1,4),and then we can have new powers of J.We will also complexify the impuritiesΦ=Φ1+iΦ2,Φ′=Φ3+iΦ4, but we will generically writeΦ.There is a simple way to see the J3/2/N behaviour.Let’s begin with the contraction (overlap amplitude)of between a2-trace operatorT r(ΦZ J1)N J1+1T r(ΦZ J2)N J2+1(x)(2.11)and the single trace operator(with J=J1+J2)lΦZ lΦZ J−lJN J+2(z)(2.12) We can see that the non-planar overlap of the two will be of orderJ2J(2.13)√where the N√√√√√J/N,down1/J from the previous.We can easily convince ourselves that all3-point non-extremal correlators in the Penrose limit will have the same fate,namely they will be subleading,and as such will not have a gravity interpretation(only away from the limit,in AdS).Moreover,exactly the same argument can be generalized easily to show that only n+m-point extremal correlators k1+...+k n=k n+1+...+k n+m survive in general.In the general extremal correlator case,with n+m=q,the closed string interaction will be of the type(again,for example,by expanding the Einstein action)g q−2s d8rdx+dx−φq−12φ(2.16) which will give a gravity amplitudeA NC∼g q−2s p(p2)q−2)q−2δ n i=1J i, m j=1J j(2.18)Nand that will be what one gets from the extremal SYM correlators as well.Theﬂat space limit of the pp wave meansµ→∞.In this limit,(p i)2=µp+n i becomes continous and the pp wave supergravity on-shell relation becomes theﬂat space one,2p+p−= p2+m2.Similarly,the string modes−2p−=µ i n i N n i (α′µp+)2(2.19)become,with the momenta(zero modes)as before(just with a change of notation), p2= iµp+N0,i,in theﬂat space limit−2p+p−= p2+M2;M2=1|x12|2k(3.1) implying a3-point function1k1k2k3<C I1C I2C I3><O(x1)O(x2)O(x3)>=N ˜α3!<˜C I1˜C I2˜C I3>As we mentioned,we can take more easily the Penrose limit in the(t, u)coordinates. Correspondingly,on the boundary we must make the conformal transformation+Wick rotation to the Lorentzian cylinder via x i=eτˆe i,τ=it.The conformal transformation acts as usual also on the boundary CFT operators,such thatO′I(τ,ˆe i)=e kτO I(x)(3.4) and if we choose−τ1≫1(x1very close the origin so that we can say|x12|∼|x2|),the 2-point function becomes<O′I2O′I1>=δI1I2e−k(τ2−τ1)(3.5) which means that the state(on the cylinder),which corresponds to the operator O(x)is O′I(τ,ˆe)|0>=e kτ|I>,where<I2|I1>=δI1I2.For the3-point function,[26]chose then to have−τ1≫1also,which is to say x1was chosen as(very close to)the origin on the plane,which is always possible.But they also chose in the case of the3-point functionτ2≫1(x2at inﬁnity),which is not always possible,since we want to keep the metric of S3×R invariant.It is possible to do that by a conformal transformation,but that takes us away from the cylinder.If one takes nevertheless alsoτ2≫1,such that|x12|≃eτ2≃|x23|,|x13|≃eτ3,one then can put the SYM3point function in a matrix element form and get<I2|O I3(τ3,ˆe3)|I1>=J1−˜k3/2˜k1!˜k2!˜k32∆cos∆[(1−iǫ)(x+−t′)](3.7)This is the propagator from the center of AdS(u=0)to the boundary(u=∞),in the pp limit.We note here already the problem.The propagator used above connects the boundary with the center of AdS,but the Penrose limit focuses only on the center of AdS,so use of (3.7)should give a zero result.If one nevertheless takes this propagator and calculates the string scattering,we will see that one still gets a result that is zero in the Penrose limit.One can use the usual AdS-CFT prescription of Witten to relate the partition func-tions Z SY M=Z string,and as the string partition function can be expressed formally and schematically asZ= DXµDhαβ...e iS= DXµDhαβ...e iS0(1+1hhαβ∂αXµ∂βXνfµν(Xρ)+...)(3.8)where fµν(Xρ)is the graviton wave function,in this case h++,and the nontrivial insertion is the vertex operator for the graviton(however,that is not of deﬁnite momentum as usual). It is also equal toS int(h++)=1δφ0|string=<I2|δiS int(h++)δφ0|SY M=<I2|O3(x3)|I1>|SY M(3.11) In the string calculation,δh++δφ0(t′,ˆe′)=R2−k I(k I+1)√N2k I/2cos k I+2[(1−iǫ)(x+−t′)]C I( y)(3.13)(note that in h MN dx M dx N only h++=h tt+hψψsurvives).Here k I=˜k3so isﬁnite in the pp limit and R2∝J,so the string amplitude is∝J1−˜k3/2/N,same as the gauge theory amplitude,thus it is notﬁnite.Theﬁnal result depends only on the boundary time t3(as theˆe3dependence was lost in the pp limit),and matches the SYM result,however it can’t be reexpressed only in pp wave(ﬁnite)quantities,since as we saw,it is actually subleading in the limit.One needs explicitly R and N.So how could we salvage this calculation and get aﬁnite amplitude in the pp wave limit? As we mentioned,we need to take O3to have large charge also.If we don’t restrict to small charge,δh++k+1(k2+∂2t)δs I k Ibut now we don’t have Y I=R−k C( y)but ratherY I(ψ, v)=[ k J ]1/22−J/2e iJψ(1−v2)J/2˜C I( v)(3.15) and since˜C( v)=R−˜k˜C( y)we getY I1≃2−k1/2(p+)˜k1/2˜k 1!f(x+,x−, y)(3.16)It is interesting to note that if we took the limit now on h++,we would get thereforeδh++N(3.17)and this is the wrong type of result:if we ignore the2−J as part of the space dependence,it is too big!(J5/2/N instead of J3/2/N).The reason for this discrepancy is rather sneaky.By takingﬁrst t′=−∞before the Wick rotation and the pp limit,we saw that in the propagator2−k cos−k(...)was replaced by e−ikt(1+u2)−k/2(see Appendix A and speciﬁcally(A.36)for details)and the interesting fact is that(apart from getting rid of the unwanted2−k factor)now,unlike before,(k2+∂2t)δs I/δφ0∼(k2+∂2t)K B∂=0inﬁrst order,so we have to look for subleading behaviour in J.Indeed,as[26]show,one getsδs I1J˜k 1!e−ip+x−−i˜k1x+−|p+|(w2+y2)/4˜C I1( y)(3.18)and now the factors leading to h++(4R2/(k+1)(k2+∂2t))don’t bring an extra J2as before, but rather J,leading to a goodh++∼J3/2which contained an operator ofﬁnite k3,therefore can’t be mapped to a string state on the pp wave(would have zero p+),and moreover it still has dependence on the boundary point, therefore the role of O I3is to couple to a boundary source,thus relating it to the3-point SYM function<I1|O(t3,ˆe3)|I2>|SY M.But in our case,<0|s I1(t, x)O I1(−∞)|0>(3.21) is mapped to<0|s I1|I1>(x+;|w|,x−, y)(3.22) which has no more dependence on the boundary point,deﬁnes a state,and depends only on x+and coordinates transverse to the boundary.It is conceivable therefore that there would be a way of constructing a vertex operator of deﬁnite momentum along x+and integrated as before over the whole space.Unfortunately,it is not clear how to calculate theﬁnite amplitude in thisﬁrst quantized string formalism,since p+changes in the amplitude(J1=J2+J3,thus p+1=p+2+p+3).It is rather a stringﬁeld theory calculation(which was already done by many people)which makes more sense.But one sees that however the calculation will be done,the result will be correct.In[26], h++has already theﬁnal leading dependence,of J1−˜k3/2/N,since the evaluation of string oscillators brings onlyﬁnite quantities,the same as the integrations.In our case,h++is already of order J3/2/N,so one just needs the correct prescription for the vertices to get the right result.4S matrices from SYMSet-upThe natural observables in AdS-CFT are correlators.But inﬂat space we have the LSZ formula that relates them to S matrices(observable),i,j d4x i e ip i x i d4y j e−ik j y j<Ω|T{φ(x1)...φ(x n)φ(y1)...φ(y m)}|Ω>( i√p2i−m2i+iǫ)( j k2j−m2+iǫ)S(p1,...p n;k1,...k m)(4.1)∼limp0i→E p i,k0j→E k jwhereiZd4xe ipx<Ω|T{φ(x)φ(0)}|Ω>=Z’s and get the S matrix!That implies a prescription for the AdS case as well.Indeed,as Giddings[8]notices, AdS-CFT takes the form<T(O( y1)...O( y2))>= Πi[d4x i K F( y i,x i)]G T(x1,...x n)(4.3)12where K F is the full multiloop bulk to boundary propagator and G T is the full amputated bulk n-point function.So if one could deﬁne the amputation process and the multiplication √by√[whereρ=√√n]1/4cos(2√2mωn=2πn1+βThenχnl(r)=A nl(R1+β)d/2−1(1+β)−l/2J l+d/2−1(E′r2E(e−iEt2E)J l+d/2−1(rE′√r d/2−1Y l m(ˆe)(4.11)That means that its Fourier transform will bel m Y l m(ˆe′)φnl m(t,r,ˆe)∝e−iEt e iE′r√E2−m2=E′√2n!R )∆(sinr2(rFlat space vs.pp wave limits for S matricesWe saw that in the global AdS parametrization,AdS-CFT relates the AdS energy E with ωnl/R,the spatial momentum direction in AdS5, k/| k|withˆe,and the AdS mass m with ∆/R.Of course that means that we have to deﬁne R and the large R limit from SYM,but from the above information we deduced2n/R=E−m,so large n is the same as large R.Notice that the above identiﬁcations are only true in the large n,large R limit.Otherwise,(∆−d4+m2R2(4.15)and extremality of correlators,∆1= i∆i meansm21R2=(n−2)(n−1)d2d2d2d2R ;˜x+=x+R;2p′+=¯∆+JR;2pψ=JµR2;˜x+=µx+⇒2p−=µ(¯∆−J);2p+=¯∆+JR,pψS5=JR2=M2+ p2transv=M2+ k2(4),AdS+ k2(4),S(4.21)We have written¯∆instead of∆since there is a change of interpretation.Here¯∆=ωnl=∆+2n+l(4.22)15takes into account both the“oﬀ-shell index”n(radial AdS oscillator number for the pp wave case)and the angular momentum l,i.e.AdS directions oscillator numbers in spherical coordinates(as we saw).On the SYM R4plane,going from an operator O to O withinsertions of D(µ1...Dµl)−traces is equivalent on the cylinder S3×R t to taking the l-th KKmode on S3,i.e.one with spherical harmonic Y l m.We could have derivatives contracted with each other,and2n corresponds to the number of contracted derivative insertions( i N0i= 2n+l in the pp wave case).Thus¯∆corresponds to taking also possible derivatives into account when counting the dimension of operators.The momentum on S5is characterized in AdS by the spherical harmonic Y l m(˜ˆe),and it corresponds in SYM on the cylinder(for global AdS)also to the(l, m)representation of operators.For Poincare AdS,the S5momentum is determined in SYM by the number of Φi insertions(the same way as Y lm(ˆe)corresponds to DµZ insertions),if they are in a large number(comparable to J).In theﬂat space limit,spherical harmonics become free waves.For example,the“spher-ical harmonic”on a circle becomesY J=e iJφ=e iJyR(4.23)For the2-sphere we have Y lm(θ,φ)and large m is as before,whereas large l is similar, as P l(cosθ)becomes cos kx.So in general,we can say that Y J m(˜ˆe)→e i˜ k· x,thus the S5 momentum is determined by Y J m,i.e.operators that are in a representation that corresponds to Y J m will have momentum k.•In conclusion,if we want to take theﬂat space limit directly,we takeﬁxed E AdS=¯∆/R.For∆∼1,n large,we get mAdS=0S matrices[8].For m AdS=∆/Rﬁxed,we get nontrivial mass and/or momenta.The sphere momentum k2S5is deﬁned bylarge J∼R∼(g2Y M N)1/4givingﬁxed pψ,and maybe large number ofΦinsertions (giving extra momenta on S5,in the directions perpendicular toψ),NΦi∼(g2N)1/4. The10d mass M is obtained by the phases e i2πnlR2+2∆R2=m2AdS+2∆R2(4.24)So n,l∼1in order to haveﬁnite k24,AdS=2∆/R2(2n+l).Thus∆∼R2,n∼1,l∼1.The sphere momentum k2S5is deﬁned by J∼R2∼(g2Y M N)1/2and by small number ofΦinsertions(of order1),that give the extra(discrete)momenta N0i.Again,the10d mass M is obtained by the phases e i2πnlThe diﬀerence between the pp wave limit and the ﬂat space limit can be understood by looking at energy of string states in the pp wave,−2p −=µ i N 0i +µ i n iJ 2= i p 2i p +(4.25)If N 0i ∼1,we get ﬁnite discrete p i ’s (“momenta”),and if J ∼g 2Y M N we get ﬁnite massM {n i }.If on the other hand,J ∼(g 2Y M N )1/4,we get p +∼1/R ∼(g 2Y M N )−1/4,or rather,we have to redeﬁne momenta to get ﬁnite results:−2p ′−=2p −(g 2Y M N )1/4+ i n i(g 2Y MN )1/4He showed then that this becomes the free(normalizable)wavefunctionC(E,R)K(E,ˆe;x)→d4ye iy(k+p) d4x′e ipx′p2(5.1)|x−y|2=One could also put y to zero by translational invariance,then obtaining the two point function minus the momentum conservationd4x e ipx p2(5.2)For the3-point function,a123f(x1,x2,x3)=we havef(p1,p2,p3)= d4x1d4x2d4x3e i(p1x1+p2x2+p3x3)f(x1,x2,x3)= d4x1e ix1(p1+p2+p3) d4x21d4x31e i(p2x21+p3x31)f(x21,x31)=δ4(p1+p2+p3)a123 d4xd4y e ip2x+ip3y|x1−x2|2k=1(cos(t1−t2)−ˆe1ˆe2)k(5.5)But[8]showed that that expression is also equal toK B(x,x′)=c dωω2nl−ω2−iǫ(5.6)whereωnl=2k+2n+l,and is actually valid even if2k is not integer and is=∆.That means that the momentum space SYM2-point function is<O I1(p1,ˆe1);O I2(p2,ˆe2)>=cδI1I2δ(p1+p2) nl m k2nl Y∗l m(ˆe1)Y l m(ˆe2)2π nl m e iωt k2nl Y∗l m(ˆe)Y l m(ˆe′)From the expression for k nl(D.3)we see that when α=0and d =4we have a Γ(−1)2in the denominator,meaning that the coeﬃcient is zero unless it’s compensated.The integral over ωgives 1/2ωnl =1/2(2n +l ).If we put n=0we get then a compensating inﬁnity and obtain − lm 10!Γ(l +2)Y ∗lm (ˆe )Y l m (ˆe ′)=−Γ(0)|x 12|2k 2...|x 1n |2k n (5.11)In the general extremal case (k 1+...+k n =k n +1+...+k n +m ),the powers are more complicated.Note that if we put (by translational invariance)x 1=0and go to S 3×R and rotate to Lorentzian signature we get<O ∗(t 1,ˆe 1)O (t 2,ˆe 2)...O (t n ,ˆe n )>=a 1...n e ik 1t 1−ik 2t 2−...−ik n t n(5.12)20For the general extremal correlator k 1+..+k n =¯k1+...+¯k m in the special case |x 1i |≪|x ij |,i =1,n ;j =1,m the powers (approximately)combine to give a similar answer:<O ∗(t 1,ˆe 1...O ∗(t n ,ˆe n )O (¯t 1,ˆ¯e 1)....O (¯t m ,ˆ¯t m )>=a 1...n ;1...m e ik 1t 1+...+ik n t n −i ¯k 1¯t 1−...−i ¯k m ¯t m (5.13)However,note that this answer is independent of ˆe i ,so cannot be correct.The point is that if we Fourier transform over the whole plane,we have the option of putting one coordinate to zero as we saw above for the 3-point function (all we miss is the overall delta function),or otherwise it gets shifted away anyway when integrating.But if we just Fourier transform over the energy,we can’t.Let us look at the extremal 3-point function (k 1=k 2+k 3)without ﬁxing any point.The 3-point function on the Lorentzian cylinder,Fourier transformed in energy isf 123(p 1,ˆe 1;p 2,ˆe 2;p 3,ˆe 3)=a 123δ(p 1+p 2+p 3) dt ′2e it ′2(p 2−k 2)(1+e −2it ′3−2e −it ′3ˆe 1ˆe 3)k 3(5.14)here as before t ′2=t 21,t ′3=t 31and we have the product of two free 2-point functions:f 123(p 1,ˆe 1;p 2,ˆe 2;p 3,ˆe 3)=a 123δ(p 1+p 2+p 3)n 2l 2 m 2k 2n 2l 2Y ∗l 2 m 2(ˆe 1)Y l 2 m 2(ˆe 2)ω2n 3l 3−p 23−iǫ(5.15)And for the general extremal n-point function we similarly getf 12..r (p 1,ˆe 1;p 2,ˆe 2;...;p r ,ˆe r )=a 12...r δ(p 1+p 2+...+p r ) n 2l 2 m 2k 2n 2l 2Y ∗l 2 m2(ˆe 1)Y l 2 m 2(ˆe 2)ω2n r l r −p 2r −iǫ(5.16)Now we will apply the Giddings procedure [8]on the general 3-point function,and then particularize for the extremal correlators.Doing the spherical harmonic integrals we getm 1m 2m 31(2l ′1+1)(2l ′2+1)(2l ′3+1)(l 1l 3l ′2;m 1m 3m ′2)(l 1l 3l ′2;000)(l 1l 2l ′3;m 1m 2m ′3)(l 1l 2l ′3;000)(l 2l 3l ′1;m 2m 3m ′1)(l 2l 3l ′1;000)(5.17)Then multiplying with p 2i −ω2n i l i and taking the limit we pick out only certain terms in the sum over n,l:l 2+l 3=l ′1,l 3−l 1=l ′2,l 1+l 2=l ′3(actually,only 2n+l is deﬁned).Then,whentaking the sum with l ′i m ′iY l ′1m ′1(ˆe 1)Y l ′2m ′2(ˆe 2)Y l ′3m ′3(ˆe 3)(5.18)we have ﬁxed l s in terms of l ′s,so we can’t use the Gaunt formula in reverse!Noticethough that one still remains with an unsaturated zero,coming from p 21−ω2n 1l 1(the rest cancel against the poles in the 3-point function).In the extremal case we get the spherical harmonic integrals (l 1=m 1=0)d ˆe 1Y ∗l 2m 2(ˆe 1)Y ∗l 3m 3(ˆe 1)Y ∗l ′1m ′1(ˆe 1) d ˆe 2Y l 2m 2(ˆe 3)Y ∗l ′3m ′3(ˆe 3) d ˆe 2Y l 3m 3(ˆe 2)Y ∗l ′2m ′2(ˆe 2)=14π 2I ηI [(∂φI )2+m 2φ2I ]+λ φ1φ2φ3(5.22)gives the two point function (see [29]and [31])<OO >=ηΓ(∆+1)∆1|x −y |2∆(5.23)and the 3-point function<O 1O 2O 3>=−λ2πd Γ(∆1−d/2)Γ(∆2−d/2)Γ(∆3−d/2)Γ(∆1+∆2+∆3−d |x −y |2α3|y −z |2α1|z −x |2α2(5.24)It is not clear by what coeﬃcient should we multiply in the general extremal case,since then it is not even clear what calculation one should do in AdS.For the extremal 3-point function though,if a 123is the coeﬃcient of the normalized 3-point function,then a 123=λb η1f 1η2f 2η3f 3(5.25)but for the S matrix we want to look only at the AdS3-point function forλ=1,i.e at the b coeﬃcient,whereas in SYM we get the a123coeﬃcient,so we should multiply the SYM result by b/a123.Putting also the C−1(E i,R)factors we get for the3-point S matrix f123(p1,ˆe1;p2,ˆe2;p3,ˆe3)=δ(p1+p2+p3)Γ(α1)Γ(α2)Γ(α3)2) {l′i m′i}k2n′2l′2k2n′3l′3C−11C−12C−13(p21−ω2n′1l′1)I{l′i,m′i}l′1m′1Y l′1m′1(ˆe1)Y l′2m′2(ˆe2)Y l′3m′3(ˆe3)(5.26) In general be will have a factor analog to b,which presumably will also have a divergent Γ(α),sof12..r(p1,ˆe1;p2,ˆe2;...;p r,ˆe r)=bδ(p1+p2+...+p r){l′i m′i}k2n′2l′2...k2n′r l′r C−11C−12...C−1r(p21−ω2n′1l′1)I{l′i,m′i}l′1m′1Y l′1m′1(ˆe1)Y l′2m′2(ˆe2)...Y l′r m′r(ˆe r)(5.27) At large R,C∼22−νΓ(∆−d/2)(−)n+l/2(2n)∆k2nl∼2ER2)2ν∼22n1)∆2+1−d(n3R2(d−1)n2(d−1)11Γ(∆2−d/2)Γ(∆3−d/2)(5.29) Now we also see how the zero in p21−ω21is cancelled,sincep21−ω21≡p21−(ω2+ω3)2∼2p1R(∆2+∆3−∆1)=4p1Rα1(5.30) and that gets cancelled becauseΓ(α1)α1=1.Putting everything together we getf123(p1,ˆe1;p2,ˆe2;p3,ˆe3)=δ(p1+p2+p3)(p2/p1)∆2+1−d(p3/p1)∆3+1−dπd R4d−5Γ(∆2)Γ(∆3)2){l′i m′i}I{l′i,m′i}l′1m′1Y l′1m′1(ˆe1)Y l′2m′2(ˆe2)Y l′3m′3(ˆe3)(5.31)Except for the numerical factors which are diﬀerent,the angular and momentum depen-dence is the same as in(B.29),except for an extra factor of p1in the denominator,and if we ﬁx the constants c=3n/2−2+ i b i and put0=d−5/2−∆i.So we still need toﬁnd a procedure to somehow get rid of the unwanted factors(in d=4)of(p2/p1)∆2−3/2(p3/p1)∆3−3/2,and of the gamma functions containg numerical∆factors.Conceivably,there should be a procedure that renormalizes the external legs such that we get the correct S matrix,as the angular momentum dependence was correct.Or maybe the problem is the fact that loop corrections modify the poles of the external propagators(as suggested by Giddings that could happen),and so maybe the residues are also modiﬁed,this giving the discrepancy that we found.One should really analyze the massive and pp waves cases and obtain the masslessﬂat space case as a limit.6ConclusionsIn this paper we have analyzed the possibility to obtain S matrices onﬂat space and pp waves from SYM.The question of pp waves is of interest in several respects.First,it is a nontrivial gravitational background,and second,we have seen that the pp wave limit already focuses in on a geodesic in the middle of AdS,so the hardest problem in the case of theﬂat space limit(getting rid of the boundary contributions)seems already solved.Of course,we need to make sure that we are indeed in the pp wave limit.For that,we have looked again at the argument that only extremal correlators survive the pp wave limit(in the same way that only large R charge operators survive it),at least as far as S matrices go.A puzzling statement in[26]that one can derive pp wave string amplitudes from nonextremal correlators was analyzed,and we showed that the amplitude is actually vanishing in the Penrose limit,and to get a nonzero result we are forced to the usual(extremal)stringﬁeld theory calculations[14,15,16,17,18,19,20],...We have deﬁned S matrices on pp waves,generalizing a procedure due to Giddings[8],ﬁrst toﬂat space with nonzero5dimensional(i.e.AdS)mass and then to pp waves.The procedure turns boundary-to-bulk propagators into normalizable wavefunctions,so we have checked that the AdS wavefunctions have the correctﬂat space and pp wave limits(and also that pp wave wavefunctions turn intoﬂat space wavefunctions).There was previously no direct test of the procedure that we are aware of.We have then tested the procedure on the correlators that we know are not renormalized: general scalar2-and3-point functions and extremal correlators.We have written down the general3-point function,but we found that it is not obvious how to proceed in the general case(it is quite complicated,and the order of limits is highly nontrivial).We have then concentrated on the extremal3-point function,the simplest case we can analyze,and also of relevance,since(with appropriate limits on the representation of operators)this correlator will survive the Penrose limit.Taking theﬂat space limit on it though,we have found a discrepancy.The AdS side result is just a delta function,that we have expressed in spherical harmonics in order to compare with the SYM result.We have found that the angular dependence works,but weget extra gamma functions containing numerical∆factors,as well as extra energy factors, (in d=4)of(p2/p1)∆2−3/2(p3/p1)∆3−3/2.We have seen that a similar thing will happen for the general extremal correlator(if we apply theﬂat space limit on it,not the pp wave limit!).So what could be the reason for the discrepancy?Of course,one answer would be to say that the procedure is not good.Basically,we are turning bulk to boundary propagators into normalizable wavefunctions.But in AdS correlators the bulk to boundary propagators are integrated over the bulk points,and theﬂat space limit supposes that we sit atﬁnite5-th coordinate r,while taking R→∞.But since we integrate over all r’s,we have to make sure that only the contribution ofﬁnite r survives.While the contribution near the boundary is negligible for normalizable wavefunctions,there is still a contribution at r∼R that is still far away from the boundary,and it is not obvious that is also small.One could maybe try to see how this aﬀects the correlator.Another potential problem was already pointed out in [8],namely that loop corrections will modify the polesωnl that we have factorized near(in LSZ fashion)by(5.10).But we have chosen especially the extremal3-point function for the certainty that the free result is exact,so at most it could be a question of deﬁning properly the limit(maybe there is some subtlety that was missed before).Aﬁnal possibility would be that theﬂat space limit does not make sense on its own, but that the pp wave limit does(and that one needs to goﬁrst in the pp wave limit,and then maybe toﬂat space).We have not analyzed the pp wave limit on the extremal3-point function(∆∼R2,n∼1,l∼1),and that could still give the right result.For the pp wave case,we know that at least theﬁrst quantized string picture(and the stringﬁeld theory calculations)work,so maybe S matrices are also OK.For those calculations,we have also seen that we can understand theﬂat space limit better as a further limit of the pp wave(see e.g.,[9,13]),so maybe the same applies here.In any case,it is clear that further work is needed to deﬁne S matrices correctly.Acknowledgements We would like to thank Juan Maldacena for pointing out to us that there is a problem with the deﬁnition of massless S matrices on the pp wave,and also to Radu Roiban for discussions.This research was supported in part by DOE grant DE-FE0291ER40688-Task A.。

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。

哈佛大学Capasso教授小组研究用界面位相突变的超材料实现超常折射的论文

Conventional optical components rely on gradual phase shifts accumulated during light propagation to shape light beams. New degrees of freedom are attained by introducing abrupt phase changes over the scale of the wavelength. A two-dimensional array of optical resonators with spatially varying phase response and sub-wavelength separation can imprint such phase discontinuities on propagating light as it traverses the interface between two media. Anomalous reflection and refraction phenomena are observed in this regime in optically thin arrays of metallic antennas on silicon with a linear phase variation along the interface, in excellent agreement with generalized laws derived from Fermat’s principle. Phase discontinuities provide great flexibility in the design of light beams as illustrated by the generation of optical vortices using planar designer metallic interfaces. The shaping of the wavefront of light by optical components such as lenses and prisms, as well as diffractive elements like gratings and holograms, relies on gradual phase changes accumulated along the optical path. This approach is generalized in transformation optics (1, 2) which utilizesmetamaterials to bend light in unusual ways, achieving suchphenomena as negative refraction, subwavelength-focusing,and cloaking (3, 4) and even to explore unusual geometries ofspace-time in the early universe (5). A new degree of freedomof controlling wavefronts can be attained by introducingabrupt phase shifts over the scale of the wavelength along theoptical path, with the propagation of light governed byFermat’s principle. The latter states that the trajectory takenbetween two points A and B by a ray of light is that of leastoptical path, ()B A n r dr ∫r , where ()n r r is the local index of refraction, and readily gives the laws of reflection and refraction between two media. In its most general form,Fermat’s principle can be stated as the principle of stationaryphase (6–8); that is, the derivative of the phase()B A d r ϕ∫r accumulated along the actual light path will be zero with respect to infinitesimal variations of the path. We show that an abrupt phase delay ()s r Φr over the scale of the wavelength can be introduced in the optical path by suitably engineering the interface between two media; ()s r Φr depends on the coordinate s r r along the interface. Then the total phase shift ()B s A r k dr Φ+⋅∫r r r will be stationary for the actual path that light takes; k r is the wavevector of the propagating light. This provides a generalization of the laws of reflection and refraction, which is applicable to a wide range of subwavelength structured interfaces between two media throughout the optical spectrum. Generalized laws of reflection and refraction. The introduction of an abrupt phase delay, denoted as phase discontinuity, at the interface between two media allows us to revisit the laws of reflection and refraction by applying Fermat’s principle. Consider an incident plane wave at an angle θi . Assuming that the two rays are infinitesimally close to the actual light path (Fig. 1), then the phase difference between them is zero ()()()s in s in 0o i i o t t kn d x d kn d x θθ+Φ+Φ−+Φ=⎡⎤⎡⎤⎣⎦⎣⎦ (1) where θt is the angle of refraction, Φ and Φ+d Φ are, respectively, the phase discontinuities at the locations where the two paths cross the interface, dx is the distance between the crossing points, n i and n t are the refractive indices of thetwo media, and k o = 2π/λo , where λo is the vacuumwavelength. If the phase gradient along the interface isdesigned to be constant, the previous equation leads to thegeneralized Snell’s law of refraction Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and RefractionNanfang Yu ,1 Patrice Genevet ,1,2 Mikhail A. Kats ,1 Francesco Aieta ,1,3 Jean-Philippe Tetienne ,1,4 Federico Capasso ,1 Zeno Gaburro 1,51School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA. 2Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, Texas 77843, USA. 3Dipartimento di Fisica e Ingegneria dei Materiali e del Territorio, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy. 4Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan and CNRS, 94235 Cachan, France. 5Dipartimento di Fisica, Università degli Studi di Trento, via Sommarive 14, 38100 Trento, Italy.o n S e p t e m b e r 1, 2011w w w .s c i e n c e m a g .o r g D o w n l o a d e d f r o m()()sin sin 2o t t i i d n n dx λθθπΦ−= (2) Equation 2 implies that the refracted ray can have an arbitrary direction, provided that a suitable constant gradient of phase discontinuity along the interface (d Φ/dx ) is introduced. Note that because of the non-zero phase gradient in this modified Snell’s law, the two angles of incidence ±θi lead to different values for the angle of refraction. As a consequence there are two possible critical angles for total internal reflection, provided that n t < n i : arcsin 2to c i i n d n n dx λθπ⎛⎞Φ=±−⎜⎟⎝⎠ (3)Similarly, for the reflected light we have ()()sin sin 2o r i i d n dx λθθπΦ−= (4) where θr is the angle of reflection. Note the nonlinear relationbetween θr and θI , which is markedly different fromconventional specular reflection. Equation 4 predicts that there is always a critical incidence angle arcsin 12o c i d n dx λθπ⎛⎞Φ′=−⎜⎟⎝⎠ (5) above which the reflected beam becomes evanescent. In the above derivation we have assumed that Φ is a continuous function of the position along the interface; thus all the incident energy is transferred into the anomalous reflection and refraction. However because experimentally we use an array of optically thin resonators with sub-wavelength separation to achieve the phase change along the interface, this discreteness implies that there are also regularly reflected and refracted beams, which follow conventional laws of reflection and refraction (i.e., d Φ/dx =0 in Eqs. 2 and 4). The separation between the resonators controls the relative amount of energy in the anomalously reflected and refracted beams. We have also assumed that the amplitudes of the scattered radiation by each resonator are identical, so that the refracted and reflected beams are plane waves. In the next section we will show by simulations, which represent numerical solutions of Maxwell’s equations, how indeed one can achieve the equal-amplitude condition and the constant phase gradient along the interface by suitable design of the resonators. Note that there is a fundamental difference between the anomalous refraction phenomena caused by phase discontinuities and those found in bulk designer metamaterials, which are caused by either negative dielectric permittivity and negative magnetic permeability or anisotropic dielectric permittivity with different signs ofpermittivity tensor components along and transverse to thesurface (3, 4).Phase response of optical antennas. The phase shift between the emitted and the incident radiation of an optical resonator changes appreciably across a resonance. By spatially tailoring the geometry of the resonators in an array and hence their frequency response, one can design the phase shift along the interface and mold the wavefront of the reflected and refracted beams in nearly arbitrary ways. The choice of the resonators is potentially wide-ranging, fromelectromagnetic cavities (9, 10), to nanoparticles clusters (11,12) and plasmonic antennas (13, 14). We concentrated on thelatter, due to their widely tailorable optical properties (15–19)and the ease of fabricating planar antennas of nanoscalethickness. The resonant nature of a rod antenna made of aperfect electric conductor is shown in Fig. 2A (20).Phase shifts covering the 0 to 2π range are needed toprovide full control of the wavefront. To achieve the requiredphase coverage while maintaining large scatteringamplitudes, we utilized the double resonance properties of V-shaped antennas, which consist of two arms of equal length h connected at one end at an angle Δ (Fig. 2B). We define twounit vectors to describe the orientation of a V-antenna: ŝalong the symmetry axis of the antenna and â perpendicular to ŝ (Fig. 2B). V-antennas support “symmetric” and “antisymmetric” modes (middle and right panels of Fig. 2B),which are excited by electric-field components along ŝ and â axes, respectively. In the symmetric mode, the current distribution in each arm approximates that of an individual straight antenna of length h (Fig. 2B middle panel), and therefore the first-order antenna resonance occurs at h ≈ λeff /2, where λeff is the effective wavelength (14). In the antisymmetric mode, the current distribution in each arm approximates that of one half of a straight antenna of length 2h (Fig. 2B right panel), and the condition for the first-order resonance of this mode is 2h ≈ λeff /2.The polarization of the scattered radiation is the same as that of the incident light when the latter is polarized along ŝ or â. For an arbitrary incident polarization, both antenna modes are excited but with substantially different amplitude and phase due to their distinctive resonance conditions. As a result, the scattered light can have a polarization different from that of the incident light. These modal properties of the V-antennas allow one to design the amplitude, phase, and polarization state of the scattered light. We chose the incident polarization to be at 45 degrees with respect to ŝ and â, so that both the symmetric and antisymmetric modes can be excited and the scattered light has a significant component polarized orthogonal to that of the incident light. Experimentally this allows us to use a polarizer to decouple the scattered light from the excitation.o n S e p t e m b e r 1, 2011w w w .s c i e n c e m a g .o r g Do w n l o a d e d f r o mAs a result of the modal properties of the V-antennas and the degrees of freedom in choosing antenna geometry (h and Δ), the cross-polarized scattered light can have a large range of phases and amplitudes for a given wavelength λo; see Figs. 2D and E for analytical calculations of the amplitude and phase response of V-antennas assumed to be made of gold rods. In Fig. 2D the blue and red dashed curves correspond to the resonance peaks of the symmetric and antisymmetric mode, respectively. We chose four antennas detuned from the resonance peaks as indicated by circles in Figs. 2D and E, which provide an incremental phase of π/4 from left to right for the cross-polarized scattered light. By simply taking the mirror structure (Fig. 2C) of an existing V-antenna (Fig. 2B), one creates a new antenna whose cross-polarized emission has an additional π phase shift. This is evident by observing that the currents leading to cross-polarized radiation are π out of phase in Figs. 2B and C. A set of eight antennas were thus created from the initial four antennas as shown in Fig. 2F. Full-wave simulations confirm that the amplitudes of the cross-polarized radiation scattered by the eight antennas are nearly equal with phases in π/4 increments (Fig. 2G).Note that a large phase coverage (~300 degrees) can also be achieved using arrays of straight antennas (fig. S3). However, to obtain the same range of phase shift their scattering amplitudes will be significantly smaller than those of V-antennas (fig. S3). As a consequence of its double resonances, the V-antenna instead allows one to design an array with phase coverage of 2π and equal, yet high, scattering amplitudes for all of the array elements, leading to anomalously refracted and reflected beams of substantially higher intensities.Experiments on anomalous reflection and refraction. We demonstrated experimentally the generalized laws of reflection and refraction using plasmonic interfaces constructed by periodically arranging the eight constituent antennas as explained in the caption of Fig. 2F. The spacing between the antennas should be sub-wavelength to provide efficient scattering and to prevent the occurrence of grating diffraction. However it should not be too small; otherwise the strong near-field coupling between neighboring antennas would perturb the designed scattering amplitudes and phases.A representative sample with the densest packing of antennas, Γ= 11 µm, is shown in Fig. 3A, where Γ is the lateral period of the antenna array. In the schematic of the experimental setup (Fig. 3B), we assume that the cross-polarized scattered light from the antennas on the left-hand side is phase delayed compared to the ones on the right. By substituting into Eq. 2 -2π/Γ for dΦ/dx and the refractive indices of silicon and air (n Si and 1) for n i and n t, we obtain the angle of refraction for the cross-polarized lightθt,٣= arcsin[n Si sin(θi) – λo/Γ] (6) Figure 3C summarizes the experimental results of theordinary and the anomalous refraction for six samples with different Γ at normal incidence. The incident polarization isalong the y-axis in Fig. 3A. The sample with the smallest Γcorresponds to the largest phase gradient and the mostefficient light scattering into the cross polarized beams. We observed that the angles of anomalous refraction agree wellwith theoretical predictions of Eq. 6 (Fig. 3C). The same peak positions were observed for normal incidence withpolarization along the x-axis in Fig. 3A (Fig. 3D). To a good approximation, we expect that the V-antennas were operating independently at the packing density used in experiments (20). The purpose of using a large antenna array (~230 µm ×230 µm) is solely to accommodate the size of the plane-wave-like excitation (beam radius ~100 µm). The periodic antenna arrangement is used here for convenience, but is notnecessary to satisfy the generalized laws of reflection and refraction. It is only necessary that the phase gradient isconstant along the plasmonic interface and that the scattering amplitudes of the antennas are all equal. The phaseincrements between nearest neighbors do not need to be constant, if one relaxes the unnecessary constraint of equal spacing between nearest antennas.Figures 4A and B show the angles of refraction and reflection, respectively, as a function of θi for both thesilicon-air interface (black curves and symbols) and the plasmonic interface (red curves and symbols) (20). In therange of θi = 0-9 degrees, the plasmonic interface exhibits “negative” refraction and reflection for the cross-polarized scattered light (schematics are shown in the lower right insetsof Figs. 4A and B). Note that the critical angle for totalinternal reflection is modified to about -8 and +27 degrees(blue arrows in Fig. 4A) for the plasmonic interface in accordance with Eq. 3 compared to ±17 degrees for thesilicon-air interface; the anomalous reflection does not exist beyond θi = -57 degrees (blue arrow in Fig. 4B).At normal incidence, the ratio of intensity R between the anomalously and ordinarily refracted beams is ~ 0.32 for the sample with Γ = 15 µm (Fig. 3C). R rises for increasingantenna packing densities (Figs. 3C and D) and increasingangles of incidence (up to R≈ 0.97 at θi = 14 degrees (fig.S1B)). Because of the experimental configuration, we are notable to determine the ratio of intensity between the reflected beams (20), but we expect comparable values.Vortex beams created by plasmonic interfaces. To demonstrate the versatility of the concept of interfacial phase discontinuities, we fabricated a plasmonic interface that isable to create a vortex beam (21, 22) upon illumination by normally incident linearly polarized light. A vortex beam hasa helicoidal (or “corkscrew-shaped”) equal-phase wavefront. Specifically, the beam has an azimuthal phase dependenceexp(i lφ) with respect to the beam axis and carries an orbitalonSeptember1,211www.sciencemag.orgDownloadedfromangular momentum of L l=h per photon (23), where the topological charge l is an integer, indicating the number of twists of the wavefront within one wavelength; h is the reduced Planck constant. These peculiar states of light are commonly generated using a spiral phase plate (24) or a computer generated hologram (25) and can be used to rotate particles (26) or to encode information in optical communication systems (27).The plasmonic interface was created by arranging the eight constituent antennas as shown in Figs. 5A and B. The interface introduces a spiral-like phase delay with respect to the planar wavefront of the incident light, thereby creating a vortex beam with l = 1. The vortex beam has an annular intensity distribution in the cross-section, as viewed in a mid-infrared camera (Fig. 5C); the dark region at the center corresponds to a phase singularity (22). The spiral wavefront of the vortex beam can be revealed by interfering the beam with a co-propagating Gaussian beam (25), producing a spiral interference pattern (Fig. 5E). The latter rotates when the path length of the Gaussian beam was changed continuously relative to that of the vortex beam (movie S1). Alternatively, the topological charge l = 1 can be identified by a dislocated interference fringe when the vortex and Gaussian beams interfere with a small angle (25) (Fig. 5G). The annular intensity distribution and the interference patterns were well reproduced in simulations (Figs. D, F, and H) by using the calculated amplitude and phase responses of the V-antennas (Figs. 2D and E).Concluding remarks. Our plasmonic interfaces, consisting of an array of V-antennas, impart abrupt phase shifts in the optical path, thus providing great flexibility in molding of the optical wavefront. This breaks the constraint of standard optical components, which rely on gradual phase accumulation along the optical path to change the wavefront of propagating light. We have derived and experimentally confirmed generalized reflection and refraction laws and studied a series of intriguing anomalous reflection and refraction phenomena that descend from the latter: arbitrary reflection and refraction angles that depend on the phase gradient along the interface, two different critical angles for total internal reflection that depend on the relative direction of the incident light with respect to the phase gradient, critical angle for the reflected light to be evanescent. We have also utilized a plasmonic interface to generate optical vortices that have a helicoidal wavefront and carry orbital angular momentum, thus demonstrating the power of phase discontinuities as a design tool of complex beams. The design strategies presented in this article allow one to tailor in an almost arbitrary way the phase and amplitude of an optical wavefront, which should have major implications for transformation optics and integrated optics. We expect that a variety of novel planar optical components such as phased antenna arrays in the optical domain, planar lenses,polarization converters, perfect absorbers, and spatial phase modulators will emerge from this approach.Antenna arrays in the microwave and millimeter-waveregion have been widely used for the shaping of reflected and transmitted beams in the so-called “reflectarrays” and “transmitarrays” (28–31). There is a connection between thatbody of work and our results in that both use abrupt phase changes associated with antenna resonances. However the generalization of the laws of reflection and refraction wepresent is made possible by the deep-subwavelengththickness of our optical antennas and their subwavelength spacing. It is this metasurface nature of the plasmonicinterface that distinguishes it from reflectarrays and transmitarrays. The last two cannot be treated as an interfacein the effective medium approximation for which one canwrite down the generalized laws, because they typicallyconsist of a double layer structure comprising a planar arrayof antennas, with lateral separation larger than the free-space wavelength, and a ground plane (in the case of reflectarrays)or another array (in the case of transmitarrays), separated by distances ranging from a fraction of to approximately one wavelength. In this case the phase along the plane of the array cannot be treated as a continuous variable. This makes it impossible to derive for example the generalized Snell’s lawin terms of a phase gradient along the interface. This generalized law along with its counterpart for reflectionapplies to the whole optical spectrum for suitable designer interfaces and it can be a guide for the design of new photonic devices.References and Notes1. J. B. Pendry, D. Schurig, D. R. Smith, “Controllingelectromagnetic fields,” Science 312, 1780 (2006).2. U. Leonhardt, “Optical conformal mapping,” Science 312,1777 (2006).3. W. Cai, V. Shalaev, Optical Metamaterials: Fundamentalsand Applications (Springer, 2009)4. N. Engheta, R. W. Ziolkowski, Metamaterials: Physics andEngineering Explorations (Wiley-IEEE Press, 2006).5. I. I Smolyaninov, E. E. Narimanov, Metric signaturetransitions in optical metamaterials. Phys. Rev. Lett.105,067402 (2010).6. S. D. Brorson, H. A. Haus, “Diffraction gratings andgeometrical optics,” J. Opt. Soc. Am. B 5, 247 (1988).7. R. P. Feynman, A. R. Hibbs, Quantum Mechanics andPath Integrals (McGraw-Hill, New York, 1965).8. E. Hecht, Optics (3rd ed.) (Addison Wesley PublishingCompany, 1997).9. H. T. Miyazaki, Y. Kurokawa, “Controlled plasmonnresonance in closed metal/insulator/metal nanocavities,”Appl. Phys. Lett. 89, 211126 (2006).onSeptember1,211www.sciencemag.orgDownloadedfrom10. D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil,“Flat dielectric grating reflectors with focusing abilities,”Nature Photon. 4, 466 (2010).11. J. A. Fan et al., “Self-assembled plasmonic nanoparticleclusters,” Science 328, 1135 (2010).12. B. Luk’yanchuk et al., “The Fano resonance in plasmonicnanostructures and metamaterials,” Nature Mater. 9, 707 (2010).13. R. D. Grober, R. J. Schoelkopf, D. E. Prober, “Opticalantenna: Towards a unity efficiency near-field opticalprobe,” Appl. Phys. Lett. 70, 1354 (1997).14. L. Novotny, N. van Hulst, “Antennas for light,” NaturePhoton. 5, 83 (2011).15. Q. Xu et al., “Fabrication of large-area patternednanostructures for optical applications by nanoskiving,”Nano Lett. 7, 2800 (2007).16. M. Sukharev, J. Sung, K. G. Spears, T. Seideman,“Optical properties of metal nanoparticles with no center of inversion symmetry: Observation of volume plasmons,”Phys. Rev. B 76, 184302 (2007).17. P. Biagioni, J. S. Huang, L. Duò, M. Finazzi, B. Hecht,“Cross resonant optical antenna,” Phys. Rev. Lett. 102,256801 (2009).18. S. Liu et al., “Double-grating-structured light microscopyusing plasmonic nanoparticle arrays,” Opt. Lett. 34, 1255 (2009).19. J. Ginn, D. Shelton, P. Krenz, B. Lail, G. Boreman,“Polarized infrared emission using frequency selectivesurfaces,” Opt. Express 18, 4557 (2010).20. Materials and methods are available as supportingmaterial on Science Online.21. J. F. Nye, M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. Lond. A. 336, 165 (1974).22. M. Padgett, J. Courtial, L. Allen, “Ligh’'s orbital angularmomentum,” Phys. Today 57, 35 (2004).23. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P.Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys.Rev. A, 45, 8185 (1992).24. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen,J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321 (1994).25. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White,“Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221 (1992).26. H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angularmomentum to absorptive particles from a laser beam witha phase singularity,” Phys. Rev. Lett. 75, 826 (1995).27. G. Gibson et al, “Free-space information transfer usinglight beams carrying orbital angular momentum,” Opt.Express 12, 5448 (2004). 28. D. M. Pozar, S. D. Targonski, H. D. Syrigos, “Design ofmillimeter wave microstrip reflectarrays,” IEEE Trans.Antennas Propag. 45, 287 (1997).29. J. A. Encinar, “Design of two-layer printed reflectarraysusing patches of variable size,” IEEE Trans. AntennasPropag. 49, 1403 (2001).30. C. G. M. Ryan et al., “A wideband transmitarray usingdual-resonant double square rings,” IEEE Trans. AntennasPropag. 58, 1486 (2010).31. P. Padilla, A. Muñoz-Acevedo, M. Sierra-Castañer, M.Sierra-Pérez, “Electronically reconfigurable transmitarrayat Ku band for microwave applications,” IEEE Trans.Antennas Propag. 58, 2571 (2010).32. H. R. Philipp, “The infrared optical properties of SiO2 andSiO2 layers on silicon,” J. Appl. Phys. 50, 1053 (1979).33. R. W. P. King, The Theory of Linear Antennas (HarvardUniversity Press, 1956).34. J. D. Jackson, Classical Electrodynamics (3rd edition)(John Wiley & Sons, Inc. 1999) pp. 665.35. E. D. Palik, Handbook of Optical Constants of Solids(Academic Press, 1998).36. I. Puscasu, D. Spencer, G. D. Boreman, “Refractive-indexand element-spacing effects on the spectral behavior ofinfrared frequency-selective surfaces,” Appl. Opt. 39,1570 (2000).37. G. W. Hanson, “On the applicability of the surfaceimpedance integral equation for optical and near infraredcopper dipole antennas,” IEEE Trans. Antennas Propag.54, 3677 (2006).38. C. R. Brewitt-Taylor, D. J. Gunton, H. D. Rees, “Planarantennas on a dielectric surface,” Electron. Lett. 17, 729(1981).39. D. B. Rutledge, M. S. Muha, “Imaging antenna arrays,”IEEE Trans. Antennas Propag. 30, 535 (1982). Acknowledgements: The authors acknowledge helpful discussion with J. Lin, R. Blanchard, and A. Belyanin. Theauthors acknowledge support from the National ScienceFoundation, Harvard Nanoscale Science and EngineeringCenter (NSEC) under contract NSF/PHY 06-46094, andthe Center for Nanoscale Systems (CNS) at HarvardUniversity. Z. G. acknowledges funding from theEuropean Communities Seventh Framework Programme(FP7/2007-2013) under grant agreement PIOF-GA-2009-235860. M.A.K. is supported by the National ScienceFoundation through a Graduate Research Fellowship.Harvard CNS is a member of the NationalNanotechnology Infrastructure Network (NNIN). TheLumerical FDTD simulations in this work were run on theOdyssey cluster supported by the Harvard Faculty of Artsand Sciences (FAS) Sciences Division ResearchComputing Group.onSeptember1,211www.sciencemag.orgDownloadedfrom。

人教版2023)必修第二册Unit 1 Culture Heritage练习含答案

人教版（2023)必修第二册Unit 1 Culture Heritage练习（含答案）单元考点巩固卷Unit 1 Culture Heritage（总分50分；时间30分钟）一、根据汉语提示拼写单词（每题1分，共10分）1. After repeated ____ (尝试), they finally succeeded.2. His second novel ____ (确立) his fame as a writer.3.Artists make a living through their(有创意的) work.4.Scientists are i the effects of diet on fighting cancer.5.Many cultural r have been preserved by the Chinese government.6.The committee rejected a(提议)for a new housing development.7. The Olympic star trained hard to(维持) his position as the world number one.8.With years of English training, I have no difficulty (组织) classes in English.9.The successful landing of Chang'e-5 on the moon further(提升) China's status as a space power.10. He designed the first "Web browser", which allowed computer users to access ____ (文件) from other computers.二、单句语法填空（每题1分，共10分）1. After leaving high school, he became a ________ (profession) football player.2. There is a front and a back entrance ________ the house.3. I was moving some furniture and I hurt my ankle ________ the process.4. I ________ (download) some introductions about some tourist attractions in the city yesterday evening.5. Only if you throw yourself into your studies will you finally achieve your long－term goal of becoming ________ archaeologist.6. One of the seven wonders of the ancient world was the ________ (pyramid)．7. It is the very one of the books that ______ (have) been translated into English.8. She doesn’t know the reason for _______ he would like to join the team.9. He ____ we (should) hold a party to welcome the new year and his ____ went through. (propose)10. You can choose to ____ someone who has hurt you even if they do not ask for ____．(forgive)三、短语填空（每题1分，共10分）in memory of, turn against, break out, in peace, defend... against... in general, used to,so that, in time,die from,1. The female hormones also help the body ____ itself ____ some kinds of infections.2. My grandfather was born in the year when the Second World War ____.3. Even those who were once for him began to ____ him.4. The monument was built ____ the dead in Wenchuan massive earthquake in 2023.5. We want to build up a harmonious society, in which everyone lives ____ and in harmony with each other.6. Many wild animals including birds have _________the forest fire in Australia.7. , _________ a learned man makes fewer mistakes in life than a man without knowledge.8. Many graduates donated their used books _______ more students can benefit from their kind behaviour.9. The cause of the power failure was found out __________ after a one-hour examination.10. My family ___________live in the city but now we have moved to the countryside.四、语篇语法填空（每题1分，共10分）Finding and keeping the right balance between progress and the 1 (protect) of cultural sites can be a big challenge. Big challenges, however,can sometimes lead 2 great solutions. In 3 1950s, the Egyptian government wanted to build a new dam across the Nile in order 4 (control) floods, produce electricity, and supply water to more 5 (farmer) in the area. But the proposal led to protests 6 water from the dam would likely damage a number of temples and destroy cultural relics. After 7 (listen) to the scientists who 8 (study) the problem, and citizens who lived near the dam, the government turned to the United Nations for help in 1959.A committee 9 (establish) to limit damage to the Egyptian buildings and prevent the loss of cultural relics. Finally, the work began in 1960. When the project ended in 1980, it was considered a great success. Not only had the countries found a path to the future that did not run over the relics of the past, but they had also learnt that it was possible for countries to work together to build a 10 (good) tomorrow.1.____2.____3.____4.____5.____6.____7.____8.____9.____ 10.____五、完成句子（每题2分，共10分）1. It looks (似乎要下雨了).2. What is known to us is (月亮绕着地球转).3. My suggestion is(我们告诉他这个结果).4. What she told you was __________________(藏在她内心深处的事情).5. This is (我父母是怎样旅行的) when they were young.单元素养检测卷Unit 1 Culture Heritage（总分100分；时间90分钟）一、阅读理解（2023届江苏省决胜新高考高三年级5月份大联考英语试题）ADown House, home of Charles Darwin:Fun factCharles Darwin, his wife, Emma, and their children lived at Down House for 40 years from 1838. Several rooms appear as if the family still live here: with croquet sets thrown into an under-stairs cupboard, a half-played game of backgammon on a side table and Emma’s knitting(毛线) left on a chair in the drawing room. Upstairs, an exhibition showcases Darwin’s voyage aboard HMS Beagle, including a reconstruction of his cabin. Outside, visitors can explore the sheltered gardens which Darwin used as an open-air laboratory, and the greenhouse in which he planted rare plants and devised botanical experiments.No room at Down House escaped Darwin’s experiments. In the drawingroom he once placed a jar of earthworms on the grand piano to see whether they could hear.Getting thereA 15-minute drive from the A21/Farnborough. Free parking. The R8 bus from Orpington stops nearby (except Sundays) or the 146 bus from Bromley North and South terminates (终点站) in Downe village, half a mile from the property. The nearest railway stations are Chelsfield or Orpington, about four miles away.Value for moneyIt’s ￡12 (adult), ￡7.20 (child), or ￡31.30 (family with 2 adults). Under 5s go free.Opening hoursOpen daily 10am-6pm from 30 March to 30 September; daily 10 a.m. — 5 p.m. between 1 to 31 October. Opening times vary through the winter (check website for details).Verdict(评价)8/10. An unstuffy educational, gentle day-trip attraction with friendly, knowledgeable staff.21. What can visitors do in Down HouseA. Play the grand piano.B. Set sail in HMS Beagle.C. Explore Darwin’s work and life.D. Try food sourced from the garden.22. How much would a couple with their 4-year-old twin sons pay foradmissionA. ￡38.40.B. ￡26.20.C. ￡31.30.D. ￡24.23. What do we know about Down HouseA. It is highly thought of.B. It charges parking fees.C. It has fixed opening hours.D. It is inconveniently located.B（2023届浙江省四校（杭州二中、温州中学、绍兴一中、金华一中）高三5月联考试题）As an intense heat wave sweeps through China, residents are seeking relief in air raid shelters and swimming pools to stay cool, and dozens of cities, including Shanghai, Chongqing and Hangzhou, have issued their highest-level red alert warnings. Shanghai has issued three red alerts this year, with the temperature hitting 40.9 Celsius on July 13, matching the record set in 2023 since 1873. The fact that Shanghai has experienced only 16 days of 40°C-plus temperatures since the city began keeping records in 1873 should give us an idea about the seriousness of the situation.Medical experts say extreme heat could cause nausea (恶心), fatigue, sunstroke and even death, with senior citizens and people with long-term illnesses particularly vulnerable to heat waves.Extreme heat events, which began a month ago, have affected the lives of more than 900 million people in China. Between June 1 and July 12,the average number of days with temperatures above 35°Cwas 5.3, up 2.4 days over normal years, breaking the national record set in 1961, according to the National Climate Center.Parts of Europe are also in the grip of heat waves and experiencing extreme weather events after the western part of North America faced extreme heat waves last year. In response to the exceptionally high temperatures, the United Kingdom has declared a national emergency and issued the highest-level red alert warning for Monday and Tuesday for the first time. More alarmingly, the average global temperature in June this year was 0.4°C higher than normal years and the highest since 1979, with temperatures in countries such as Spain, France and Italy exceeding 40°C.Unfortunately, extreme heat, which is directly related to climate change, will become more frequent and intense in the next 30 years, setting new records for high temperatures. As global warming intensifies, losses and devastation will increase, forcing natural and human systems to raise their adaptation limits.28. What can we know about the heat waves this yearA. It may cause more harm to the old and people with long-term illnesses.B. The number of days above 35°Cin June breaks the national record.C. Shanghai has experienced a higher temperature than that in 2023.D. The whole Europe as well as America are suffering from the heat waves.29. How does the author develop the textA. By analyzing and concluding.B. By explaining and contrasting.C. By giving examples and quoting.D. By giving figures and comparing.30. According to the writer, what is the trend of extreme heatA. Becoming more serious.B. Remaining stable.C. Staying unpredictable.D. Getting controllable.31. What is the text mainly aboutA. The solutions to the climate change.B. The economic losses from heat waves.C. The increase of severe heat waves.D. The destructive effect of global warming.二、七选五（2023届广州市科学城中学高三第二学期英语5月月考试卷）Color psychology is the study of how colors affect human behavior, mood, or physiological processes. Colors affect our feelings and memories. ___16___ Companies choose colors that they believe will motivate customers to buy their products and improve brand awareness.Color perception is very subjective, as different people have different ideas about and responses to colors. Several factors influence color perception, which makes it difficult to determine if color alone impactsour emotions and actions.___17___ In some cultures, for example, white is associated with happiness and purity. In a situation where a woman is wearing a white wedding dress, is she happy because she is influenced by the color white or because she is getting married To someone from a different culture, wearing white may signify sadness. ___18___While no direct cause and effect relationship between color and behavior has been found, some generalizations about colors and what they may symbolize have been determined. Colors including red, yellow, and orange are considered warm colors. ___19___ Cool colors include blue, violet, and green. These colors are associated with calmness and coolness.___20___ We see colors with our brains. Our eyes are important for detecting and responding to light, but it is the brain's visual center in the occipital lobes（枕叶）that processes visual information and assigns color. The colors we see are determined by the wavelength of light that is reflected. The brain integrates these wavelength signals enabling us to distinguish among millions of different colors.A. We don't actually see colors with our eyes.B. Our brain associates the wavelength with a color.C. Colors also have been used to treat various diseases.D. They are even thought to influence our buying choices.E. These colors are thought to stimulate exciting emotions.F. The influential factors of color perception include age and culture.G. This is because white is associated with sorrow and death in those cultures.三、完形填空（2023届山东省威海市高三5月高考模拟考试（二模）英语试题）Jim Quick grew up on a farm in Orchard Hill, Georgia. There he learned how___21___bees are to the food we eat. When he was 11, his grandfather asked him to start a beehive to keep the___22___growing strong.“Forty-two years later, and I’m still___23___,”Quick said in a recent interview. Before his retirement, the master beekeeper taught entomology (the study of insects) at the University of Georgia in Athens. Quick is still learning about bees on the farm where he grew up. But he is worried: The bee___24___is in danger. Nearly 40% of beehives in the United States were___25___in 2023 alone. This___26___Quick and other scientists because bees are essential to food___27___.They pollinate (授粉) flowering crops and plants, ___28___the production of seeds and fruits.The declining bee population can be___29___by several factors, including parasites (寄生物) that kill or weaken the hives. Habitat loss,____30____, and climate change are also to blame. Pesticides get into the nectar (花蜜) of plants, which is essential to a bee’s____31____.Poisonous substances in the nectar harm the bees’ memory.Scientists are searching for____32____to help the bee population. But Quick already has a(n)____33____.“Set up a beehive at school,” he said, “and learn more about pollinators.” Kids aren’t too young to____34____. After all, just at the age of 11 Quick____35____his first beehive. “We could grow more bees,” he said, “if we had more people to do it.”21. A. vital B. friendly C. addicted D. sensitive22. A. bees B. bushes C. crops D. roots23. A. trying B. working C. learning D. teaching24. A. growth B. migration C. diversity D. population25. A. lost B. moved C. collected D. discovered26. A. disappoints B. concerns C. shocks D. challenges27. A. safety B. storage C. flavor D. production28. A. relying on B. resulting in C. speeding up D. controlling over29. A. caused B. identified C. worsened D. limited30. A. enemies B. disasters C. diseases D. chemicals31. A. size B. life C. diet D. memory32. A. clues B. solutions C. volunteers D. donations33. A. job B. idea C. choice D. schedule34. A. help B. remember C. decide D. communicate35. A. decorated B. designed C. started D. bought四、语法填空（2023届湖南省长沙市明德中学5月高考全仿真模拟英语试卷）阅读下面短文，在空白处填入1个适当单词或括号内单词的正确形式。

关于男生女生身体语言差异的英语作文看法

关于男生女生身体语言差异的英语作文看法全文共5篇示例，供读者参考篇1Boys and Girls Talk With Their BodiesHave you ever noticed how boys and girls seem to move and act differently? Even though we're all kids, there are some big differences in how we use our bodies to communicate without words. It's pretty interesting if you pay attention!Let's start with the boys. A lot of times, boys take up more space. They stretch out their arms and legs, sprawling all over their desks and chairs. They might slouch down or lean way back. Some boys like to roughhouse and playfully shove each other around. When they get excited about something, they might punch the air or make big movements with their whole body.Boys also tend to have a heavier stride when they walk - it's like their feet are stomping instead of just stepping. Their hand motions are bigger too, waving their arms around animatedly when they're telling a story or explaining something. I've seen some boys put their hands on their hips in a confident stance,puffing out their chests a little. It's almost like they're trying to make themselves look bigger and stronger.Girls, on the other hand, often seem more compact and contained in how they hold themselves. They tend to sit with their legs together or crossed, keeping their elbows tucked in close to their bodies. Instead of big, sweeping arm motions, girls usually gesture with smaller, more delicate hand movements when they talk.Girls also tend to lean in towards each other more, maintaining pretty close proximity. They might touch each other's arms lightly while conversing or stand Side-by-side with their shoulders brushing together. It's like they're creating a tighter, more intimate personal space bubble around them.I notice that a lot of girls play with their hair or smooth down their clothes while talking - it's almost like they're preening or primping a little. They'll tilt their heads to the side or bob their knees up and down in a cutesy way. Some girls accentuate what they're saying by batting their eyelashes or pouting out their lips. Boys just don't really do that kind of stuff!Of course, these are just generalizations based on what I've noticed. There are always exceptions - I know some girls who are more physical and animated, just like how some boys are morecontained. But in general, it seems like boys take up more space and move more dynamically, while girls tend to be more compact and do smaller, softer gestures.It's funny to think that we communicate so much without even opening our mouths, just through our posture, movements, and subtle little mannerisms. I wonder if we learn these body language styles from watching others around us, or if some of it is just natural instincts based on if you're a boy or a girl? Figuring out what all these unspoken signals and cues really mean is something even adults are still studying and debating about.But one thing's for sure - paying closer attention to body language has made me way more aware of the tiny details in how we express ourselves physically. I like trying to interpret what my friends' gestures and posture might be communicating. And who knows, maybe understanding these differences better will help boys and girls communicate more clearly, even before any words are spoken!篇2Body Talk: The Secret Signals of Boys and GirlsHave you ever noticed how boys and girls seem to speak a totally different language sometimes? I'm not talking aboutEnglish or Spanish or any of those languages we learn in school. I mean the secret body language that we all use without even thinking about it!Body language is like a hidden code that tells you how someone is really feeling, even if their words say something different. And let me tell you, the body language of boys and girls can be worlds apart. It's like we're from two different planets sometimes!Let's start with the basics – how we walk and stand. Boys tend to take up a lot of space when they move around. Their steps are bigger and their arms swing more. It's like they're trying to say "I'm here, watch out world!" Girls, on the other hand, often walk and stand in a more closed-off way, with their arms tucked in and their steps smaller. It's almost like they're trying to make themselves seem smaller and less noticeable.Then there are the hands and arms. Boys love to use big gestures when they talk, waving their arms all over the place. Sometimes they'll even playfully shove each other around a bit. But girls are much more likely to use small, precise hand movements, and they hardly ever make physical contact like that.And let's not forget about eye contact! Boys tend to look away a lot when they're talking, like they're scanning the roomfor something more interesting. But girls? They'll look right at you, holding your gaze like they're trying to catch every single word you say.Of course, these are just general patterns – not every boy and girl fits into these boxes perfectly. Some girls can be just as bold and loud as any boy, and some boys can be shy and reserved. But in general, these body language differences do seem to pop up a lot.So why do boys and girls communicate so differently with their bodies? Well, some experts think it might have to do with how we're raised. From a very young age, boys are often encouraged to be more physically active and assertive, while girls are taught to be more nurturing and socially aware. These messages can get baked right into our body language without us even realizing it.Other researchers believe the differences could be biological. Like, maybe girls are naturally more tuned into subtle social cues because they needed to be able to read situations well to take care of babies back in ancient times. And maybe boys are more physically demonstrative because they had to compete and stand their ground to protect their families and territories.Whatever the reasons, learning to decode these secret body signals can be really useful. For example, if a girl is looking down and playing with her hair, she might be feeling shy or nervous. But if a boy is pacing around and not making eye contact, he could be bored or restless instead. Once you start to pick up on these clues, you can respond in a way that makes both of you feel more comfortable.Of course, body language is just one piece of the communication puzzle. The words people actually say are important too, as well as the tone of their voice and other context clues. But being a master code-breaker of all the gestures and movements we make can go a long way towards really understanding each other.So the next time you're talking to someone, keep an eye on their body language as well as their words. Those secret physical signals just might give you a surprising new window into how they're feeling. In the boy/girl body talk battle, a little awareness can go a long way!篇3Body Language Differences Between Boys and GirlsHave you ever noticed how boys and girls move differently? Even when we're just sitting in class, you can see the differences in our body language. As a kid, I've spent a lot of time observing my classmates and I've picked up on some interesting patterns!Let's start with the way we sit. A lot of the boys in my class slouch down in their chairs or sprawl out with their legs wide open. They seem to take up a lot of space without even realizing it. The girls, on the other hand, tend to sit up straighter with their legs together or crossed. We're always being told to have good posture, so maybe that's why.When boys and girls walk down the hallway, you'll notice another big difference. Most of the boys just stroll along without really paying attention to their movements. They might shove each other playfully as they go. But a lot of the girls in my grade walk with their arms tight by their sides and look straight ahead. I think we're just trying to avoid bumping into anyone or drawing too much attention.Body language also shows up a lot when we're talking to our friends. The boys are usually louder and make bigger hand gestures, like they're telling an exciting story. Sometimes they'll even stand up or move around as they speak. But when girls arechatting, our hand motions are smaller and we stay seated. We also make more eye contact and lean in closer while listening.During recess, keep an eye on the kids playing sports or running around. The boys tend to take up the whole field or playground with their wild movements. They'll tackle each other, wave their arms, and just use their whole bodies. Girls engage in less rough physical activity at recess. We might skip rope, do cartwheels, or just hang out talking in tighter groups.Even our facial expressions give away differences in body language between boys and girls. Boys tend to look angrier or more intense, with furrowed brows and clenched jaws. They're not afraid to let their emotions show on their faces. Girls, however, more often have smiley, open expressions or look interested and attentive.Now this isn't true for absolutely every boy and girl - we're all individuals, after all. And as we get older, maybe these body language patterns will change. But for now in elementary school, I've definitely noticed distinct differences in how the two genders move, gesture, and express themselves through their bodies.It's kind of funny to pay such close attention to these little details. But I think it's an interesting window into how boys and girls are socialized from a young age to behave differently, evenwithout realizing it. Our body language gives away subtle cues about our personalities, emotions, and comfort levels.Just something to keep an eye out for the next time you're on the playground or in the classroom! You might start to recognize those telling signs of "boy body language" versus "girl body language." And who knows, maybe understanding these differences can help boys and girls communicate and get along better. At the very least, it's an amusing way to pass the time during a boring lecture!篇4Boys and Girls and Their Body Language: What I've NoticedHave you ever paid close attention to how boys and girls move and act differently? Even though we're all kids, I've seen some interesting differences in how the boys and girls in my class use their bodies to communicate without words. Body language is like a silent language that people use without even realizing it sometimes!The biggest difference I've noticed is that the girls in my class tend to face each other more directly when they're talking and laughing together. They'll stand or sit facing each other, making lots of eye contact. The boys, on the other hand, tend toface in the same direction side-by-side instead of straight on. I've seen groups of boys all facing the same way, almost forming a line as they goof around together.I've also noticed that girls tend to be more expressive with their faces and hands when communicating. They smile more, raise their eyebrows, and use gestures like waving their hands around to help get their point across. The boys don't make as many animated facial expressions usually. But the boys do tend to use bigger body movements like jumping up and down or punching the air when they get excited about something.Another cool thing is that girls seem to feel more comfortable standing or sitting quite close together when they talk. My guy friends and I usually give each other a bit more personal space, keeping a wider radius around us. Maybe that's because girls are just generally more touchy-feely and affectionate with their friends through hugs, hand-holding, and that kind of thing.One difference that kind of bugs me is how much the girls fidget and play with their hair, jewelry, or clothes while they're talking. It's almost like they can't keep still! The boys generally just stand or sit normally without fiddling with anything. I guess we don't have as many accessories to twiddle.When it comes to disagreeing with each other, I've seen some differences too. If two girls have a disagreement or argument, there's a lot more dramatic body language involved –stamping feet, hands on hips, rolling eyes, and tossing their hair around. With two guys disagreeing, there's more challenging body posture like puffing out their chests or invading each other's personal space by getting right in each other's faces.Those are some of the biggest differences in body language I've noticed between boys and girls in my grade so far. It's almost like we're sending different physical signals without meaning to! I'm sure there are exceptions and not every boy or girl fits these generalizations. But for the most part, these unspoken cues do seem to be a real thing based on what I've seen. Body language is a pretty fascinating secret language when you start paying attention to it!篇5Boys and Girls Are Really Different!Have you ever noticed how boys and girls seem to move and act in totally different ways? I've been watching my classmates closely and I've seen some really big differences in how the boys and girls use their bodies to communicate without even sayinganything! It's kind of like they're speaking two different languages with their bodies.Let me tell you about what I've observed. The boys in my class are always moving around a lot. They can't seem to sit still for very long. They're constantly fidgeting, tapping their feet, playing with something in their hands, or getting up to walk around the room. Their body movements are really big and exaggerated too. When they talk, they use huge arm movements and hand gestures. They take up a lot of space with how much they move their arms and legs out to the sides. It's like they're trying to make themselves look bigger or something.The girls, on the other hand, are much more still and contained with their bodies. They can sit nicely and pay attention for way longer than the boys can. When they do move around, it's with smaller, more delicate movements. Their hand gestures when talking are pretty small and close to their bodies. Instead of taking up a lot of space, they tend to keep their arms and legs pulled in close. The girls face each other more directly too when they talk, while the boys are always turning their bodies away at angles.I've noticed the girls tend to lean in towards each other and make more eye contact, like they're really paying close attention.The boys hardly ever make eye contact and are always looking around the room instead of at who they're talking to. The girls also smile, laugh, and touch each other's arms way more than the boys do when conversing.The way we use our faces to express emotions is really different between boys and girls too. The girls in my class make lots of different facial expressions all the time based on how they're feeling. If something's funny, they scrunch up their faces in big smiles. If they're confused, they make these cute frowny faces. The boys, on the other hand, mostly just have one expression, which looks kind of mad or frowning, even if they're not. It's harder to tell what the boys are feeling based on their faces.You'd think the boys would be sloppier with how physical and wild they are with their movements, but it's actually the opposite! I've seen the boys shake hands and greet each other following the same routines every time, like it's a secret handshake or dance they've practiced. The way they sit or stand seems very precise, like they have to hold perfect posture. But the girls are way more relaxed and comfortable looking in how they arrange their bodies. I guess the boys have to follow certainrules for their body language while the girls can just move however feels natural.There are a couple boys in our class who don't follow these typical "boy" body language patterns though. They're quieter and make more eye contact and small movements like the girls. And there are some girls who are rowdier and more physical like the boys. But for the most part, you can really see the differences between how the boys and girls express themselves through their bodies and postures and gestures and facial expressions. It's like they're speaking totally different body languages!I wonder if these differences come from how kids are raised from a young age? Like, are boys encouraged to be more physical and take up space while girls are taught to be more delicate and contained? Or maybe it's just biological and deeply ingrained in how our brains and bodies develop differently? There could also be cultural influences on what body language is seen as appropriate for boys versus girls in our society. Whatever the reasons, it's a really fascinating thing to observe in my class every day. Body language is like a secret code and it's so interesting to see the differences in how boys and girls use it. I'm going to keep watching my classmates to try to crack the code even more!。

PPV的合成和光致发光光谱

收稿日期:1999-07-19基金项目:国家自然科学基金重大课题(29992530)第21卷第2期半　导　体　光　电Vol.21No.22000年4月Semiconductor OptoelectronicsApr.2000文章编号:1001-5868(2000)02-0129-03PPV 的合成和光致发光光谱莫越奇1,贾德民1,汪河洲2,郑锡光2(1.华南理工大学材料学院,广州510641;2.中山大学超快激光光谱国家重点实验室,广州510275)摘　要:　用Lenz 法和Burn 法合成了PPV ,发现热转化温度是改变光致发光光谱的主要因素,随着热转化温度升高,谱峰发生红移,强度下降,谱峰形状变化很大;轻微的氧化、膜厚对发光强度有影响,对谱峰位置、形状影响很小;另外渗析不完全会使Lenz 法合成的PPV 谱峰发生蓝移和形状变化。

关键词:　PPV ;发光二极管;光致发光中图分类号:　TN304.52 文献标识码:　ASynthesis of Poly(Phenylene Vinylene)s and Measurementsof Photoluminescence SpectraMO Yue 2qi 1,J IA De 2min 1,WAN G He 2zhou 2,ZHEN G Xi 2guang 2(1.F aculty of Material Science and E ngineering ,South China U niversity of T echnology ,G u angzhou 510641,China ;2.State K ey Lab.of U ltra -fast Laser Spectrascopy ,Zhongshan U niversity ,G u angzhou 510275,China)Abstract :　Poly (phenylene vinylene )s are synthesized by utilizing Lenz and Burn ’s methods ,re 2spectively.It is found that the thermal converting temperature is the key factor affecting the photolu 2minescence (PL )spectra.That is to say ,as converting temperature increases ,red -shift appears ,while PL intensity decreases and shape changes .Slight oxidization and the thickness of the film have an ef 2fect on the PL intensity of PPVs and the blue -shift and spectra shape change will occur unless the prepolymer synthesized by Lenz ’s method is thoroughly dialyzed.K eyw ords :　poly (phenylene vinylene );light -emitting diodes ;photoluminescence1　引言1990年Brroughes 等人[1],用聚苯乙炔(PPV )发光材料制成了第一只聚合物发光二极管,经过对材料和器件的多次重大的改进之后,电致发光效率已由最早的0.01%提高到2%[2],接近于应用水平。

基于含可变因子广义S变换的瑞雷面波频散曲线提取方法

基于含可变因子广义S变换的瑞雷面波频散曲线提取方法周竹生;杨朋凯;陈友良【期刊名称】《物探化探计算技术》【年(卷),期】2012(034)005【摘要】The key step of surface wave data processing is to extract its dispersion curve. As the frequency resolution and time resolution can not be taken into account by S-transform or the generalized S-transform for the whole time samples simultaneously, a new method by using the generalized S-transform with variable-factors to extract dispersion cure of transient Rayleigh wave is present. With the variable-factors, the width of gaussian window versus frequency varies as expected, rather than simply narrows as frequency increasing. This method can improve the frequency resolution and the time resolution of the local bands with purpose, especially the low and high frequency bands. The results of theoretical model testing and real data trials show that the method of surface wave dispersion curve extraction based on generalized S-transform with variable-factors is feasible and practical.%面波频散曲线的提取是面波资料处理的关键.鉴于S变换和广义S变换不能全时段兼顾频率分辨率和时间分辨率的缺点,提出了含可变因子广义S变换进行瑞雷面波频散曲线的提取方法.可变因子的引入使得高斯窗函数的宽度随频率发生变化时具有目的性,而不是简单地随着频率的增大而变窄.该方法可以有针对性地改善局部频段(特别是低频段和高频段)的频率分辨率及时间分辨率.通过理论模型和实际资料试算表明:含可变因子广义S变换提取面波频散曲线的方法具有可行性和实用性.【总页数】5页(P518-522)【作者】周竹生;杨朋凯;陈友良【作者单位】中南大学地球科学与信息物理学院,湖南长沙410083;中南大学地球科学与信息物理学院,湖南长沙410083;中南大学地球科学与信息物理学院,湖南长沙410083【正文语种】中文【中图分类】P631.4【相关文献】1.含层状液体介质瑞雷面波频散曲线正演分析 [J], 李长征;肖柏勋2.基于τ-p变换的瑞雷面波频散曲线提取方法研究 [J], 熊治涛;唐新功;陈义群3.基于瑞雷面波频散曲线反演的岩体力学参数求取方法 [J], 潘自强;徐贵来;吴曲波4.基于多模式的多重滤波方法提取瑞雷面波频散曲线 [J], 陈杰;熊章强;张大洲;张俊;章游斌5.基于相位差法和多重滤波法结合的瑞雷面波频散曲线研究 [J], 谌强;熊章强;张大洲因版权原因，仅展示原文概要，查看原文内容请购买。

不同形状粗糙元在诱导超声速边界层转捩中的应用

不同形状粗糙元在诱导超声速边界层转捩中的应用ZHOU Yunlong;LIU Wei;WU Dong【摘要】为了研究不同形状粗糙元诱导边界层转捩机理的差异,采用五阶精度加权紧致非线性格式数值模拟了Ma=4.20条件下方柱形、圆柱形、钻石形和半球形粗糙元引起的超声速平板边界层转捩问题.结果表明:方柱形粗糙元产生的分离区长度较大,分离区中存在较强的非定常扰动并产生绝对不稳定机制使边界层很早就发生转捩;钻石形粗糙元分离区的展向宽度较宽,导致分离区中的涡结构向下游发展时形成的湍流尾迹区较宽;而圆柱形和半球形粗糙元诱导边界层转捩的能力相对较弱.【期刊名称】《国防科技大学学报》【年(卷),期】2018(040)006【总页数】6页(P17-22)【关键词】超声速;边界层转捩;粗糙元形状;数值模拟【作者】ZHOU Yunlong;LIU Wei;WU Dong【作者单位】;;【正文语种】中文【中图分类】V211.3由于湍流流动会产生明显高于层流流动的摩阻系数和传热系数，直接影响超声速和高超声速飞行器设计，因此合理控制边界层转捩意义重大。

通常边界层转捩控制出于两种不同的目的：一种是延迟转捩，比如对于再入大气层的宇宙飞船转捩推迟可以减少热量的传入；另外一种是促进转捩，例如在高超声速超燃冲压发动机进气道设计中，为了避免流动分离导致的进气道入口壅塞，需要保证进气道入口处的流动为湍流，通常在飞行器前体表面安装强制转捩装置诱导边界层转捩[1]。

粗糙元作为工程中常用的一种涡流发生器，可以有效地诱导边界层转捩，大量的风洞试验和数值模拟[2-3]进行了粗糙元诱导边界层转捩机理的研究。

其中不同形状粗糙元诱导边界层转捩的机理和效果不同，因此研究粗糙元形状对边界层转捩的影响在工程应用中具有重要的指导意义。

早在1969年，Whitehead[4]试验对比了高超声速条件下不同形状粗糙元的流动结构和阻力问题，包括半球形、三棱柱形、圆柱形和两个楔形的涡流发生器外形，结果表明楔形涡流发生器对应着最小的流场阻力。

基于弹性力学的超构材料_阮居祺

摘要
近年来以微结构为基本构造单元的人工超构材料, 由于具有自然材料所不具备的
可设计的奇异物性, 在材料学、声学、光学、电磁学以及信息能源等领域具有巨大的发展潜力. 超构材料的研究脱胎于电磁超构材料, 但是近年来在声学、热学、静电、静磁学以及弹性力学领域取得了飞跃的发展, 大大拓展了超构材料的研究领域. 借助 Milton 图重点阐述了基于弹性力学的新型超构材料的超常特性及其主要类别: 例如具有负的质量密度和负弹性模量的声学超构材料, 具有负泊松比的拉胀超构材料, 具有剪切模量 G=0 的反胀超构材料, 以及高强度的超轻材料等新奇的人工超构材料. 不仅如此, 还结合变换力学着重描述了声波和弹性波在这类弹性力学超构材料中的传播特性, 以及详细阐述了负弹性参数超构材料界面的声表面波的特征及其物理效应 . 最后结合弹性力学超构材料在我国的研究现状 , 对利用弹性力学超构材料和声波超构材料操纵弹性波和声波的传播以及开发设计新型弹性力学超构材料等问题作了总结与展望, 希望推进此类材料在诸多研究领域的应用.
阮居祺等: 基于弹性力学的超构材料
材料这一概念应当拓展到包含声子晶体在内的所有具有奇异性能的人工复合结构材料[3]. 声学超构材料可展现出许多奇异的物理现象和效应 , 如声波的低频带隙[4]、负折射[5]、隐身[6]等. 所谓基于弹性力学的超构材料泛指具有超常规力学性能的一类人工复合材料 , 该类材料的弹性力学性能不仅受限于其微结构组分, 还强烈依赖于它的结构. 因此基于弹性力学的超构材料包含声学超构材料 , 它能展示出均匀材料不具备的奇异弹性力学性能: 如负密度、负弹性模量、各向异性密度张量、负泊松比等. 这类材料将在材料学、声学、结构力学、建筑学、地震学等学科领域具有重要研究价值, 同时在新型材料、信息、能源技术方面会有重要应用. 特别要注意的是, 由于服役环境和条件不同 , 针对弹性力学超构材料的研究将不仅涉及静态的弹性力学问题 , 也会涉及流体中的声波传播和弹性体中固体弹性波的传播问题.

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

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。