Quadratic Vectors Research Articles

Misalignment algorithm of a wide-field survey telescope based on third-order quadratic nodal aberration theory

Off-axis three-mirror anastigmatic (TMA) telescopes with designed decenters and tilts, have many misalignment degrees of freedom and strong coupling between each misalignment degree of freedom. Therefore, it is difficult to establish misalignment equations only using A222 and A131 in nodal aberration theory (NAT). In addition, for off-axis TMA optical systems with designed decenters and tilts, the robustness of the existing fifth-order NAT misalignment calculation algorithm based on high-order Zernike coefficients and boresight errors decreases, so it is difficult to realize its engineering application. To solve the issue of insufficient practicality of the existing misalignment algorithm based on fifth-order NAT, a third-order NAT calculation algorithm based on quadratic aberration field decenter vectors is derived and established. Two concepts of inherent aberration field decenter vector and misalignment aberration field decenter vector are proposed. Taking an off-axis TMA optical system with 6-m focal length as the research object, simulations, and alignment verification experiments were carried out. Compared with the existing fifth-order NAT misalignment algorithm, the results show that when measurement noise is not considered, the two methods can both obtain convergent calculation results, and the average RMS wavefront errors (WFE) of the optical system are both corrected to be below 0.0574 waves. When different levels of measurement noise are introduced, the robustness of the fifth-order NAT misalignment algorithm decreases, and there are even cases where the optical system completely fails to be corrected. However, the algorithm based on quadratic aberration field decenter vectors shows better robustness. Under different levels of measurement noise, this algorithm could correct the average RMS WFE of the optical system to around 0.0574 waves.

Convergence characteristics of iterative learning control for discrete-time singular systems

This paper investigates the convergence characteristics of the conventional P-type iterative learning control (ILC) scheme and exploits a gain-adaptive iterative learning control mechanism for a class of linear discrete-time singular systems. Based on the lifted vector technique, the paper reforms the discrete-time singular system as a kind of algebraic input-output transmission. For the conventional P-type ILC scheme, the asymptotical convergence in terms of the tracking-error vector is achieved and the monotonic convergence in the sense of 2-norm of the tracking-error vector is derived. Further, in order to improve the learning performance, a gain-adaptive iterative learning control (GAILC) strategy is developed, which argues the iteration-time-variable gain vector while minimising the increment of quadratic tracking-error vectors of two adjacent iterations. The existence of the optimal gain vector is explored through the optimisation criterion and the algebraic approach of the columns/ rows exchanging transformation of matrix. Then the non-conditionally strictly monotonic convergence of the GAILC is made by studying the eigenvalues of the quadratic function. Finally, the validity and the effectiveness of the P-type ILC are numerically demonstrated and the remarkable outcomes of the GAILC are illustrated.

Fast reconstruction of the aberrated scanning lithographic image by a quadratic imaging model and an integral transfer function.

Since the pupil function is defined as a time-invariant system, the traditional partially coherent imaging model is time-consuming to calculate the effect of spatially varying wavefront aberrations on the scanning image. A fast reconstruction method of the aberrated scanning aerial image is presented for the scanning projection lithographic tool. In the proposed method, the principal components (PCs) are used to decompose and reconstruct the aberrated aerial image. Due to the exact quadratic relationship between the PC coefficients and the Zernike coefficients, the integration of the PCs in the intensity domain can be transformed into the integration of quadratic Zernike vectors when reconstructing the scanning aerial image. An integral transfer function is introduced to describe this process. This method can not only reconstruct the aberrated scanning image quickly but can also obtain the explicit relationship between the Zernike coefficients and the aberrated scanning aerial image.

On the one-mode quadratic Weyl operators

It has been known, in [7], that one-mode quadratic Weyl operators are well-defined as unitary operators acting on the quadratic Fock space. These operators were defined by their action on the finite particles space. However, their action on the domain of the quadratic exponential vectors is still unknown. In this paper, we provide a new reformulation (w.r.t. [1]) of the quadratic exponential vectors and we compute the action of the one-mode quadratic Weyl operators on the set of these exponential vectors. Then, we prove the independence of the one-mode quadratic Weyl operators parameterized by the principal domain associated with the quadratic Heisenberg group obtained in [7]. This significant contribution to the program of developing the quadratic white noise calculus constitutes a step toward the C⁎-representation of the renormalized square of white noise algebra.

Image Feature Matching Based on SIFT Algorithm

In the process of image matching, it is involved such as image rotation, scale zooming, brightness change and other problems. In order to improve the precision of image matching, image matching algorithm based on SIFT feature point is proposed. First, the method of generating scale space is introduced. Then, the scale and position of feature points are determined through three dimension quadratic function and feature vectors are determined through gradient distribution characteristic of neighborhood pixels of feature points. Finally, feature matching is completed based on the Euclidean distance. The experiment result indicates that using SIFT feature point can achieve image matching effectively.

Locking-free compressible quadrilateral finite elements: Poisson’s ratio-dependent vector interpolants

In any high-quality viscoplastic calculation, the objectivity related to shape and constitutive parameter indifference is desirable, especially when a continuous deformation into concavity is tracked. In the physical x−y frame, quadratic displacement vectors—explicitly coupled via Poisson’s ratio—encompass bar and Euler–Bernoulli beam modes. Analytically, the unique set of shape functions yields nodal loads and the element stiffness matrix from its area and the first and second area moments. Closed-form Mathematica results exhibit full compliance with all requirements of Iron’s patch tests.

A Generalized Component Mode Synthesis Approach for Flexible Multibody Systems With a Constant Mass Matrix

A standard technique to reduce the system size of flexible multibody systems is the component mode synthesis. Selected mode shapes are used to approximate the flexible deformation of each single body numerically. Conventionally, the (small) flexible deformation is added relatively to a body-local reference frame which results in the floating frame of reference formulation (FFRF). The coupling between large rigid body motion and small relative deformation is nonlinear, which leads to computationally expensive nonconstant mass matrices and quadratic velocity vectors. In the present work, the total (absolute) displacements are directly approximated by means of global (inertial) mode shapes, without a splitting into rigid body motion and superimposed flexible deformation. As the main advantage of the proposed method, the mass matrix is constant, the quadratic velocity vector vanishes, and the stiffness matrix is a co-rotated constant matrix. Numerical experiments show the equivalence of the proposed method to the FFRF approach.

Improved constraint consensus methods for seeking feasibility in nonlinear programs

The Constraint Consensus method moves quickly from an initial infeasible point to a point that is close to feasibility for a set of nonlinear constraints. It is a useful first step prior to launching an expensive local solver, improving the probability that the local solver will find a solution and the speed with which it is found. The two main ingredients are the method for calculating the feasibility vector for each violated constraint (the estimated vector to the closest point that satisfies the constraint), and the method of combining the feasibility vectors into a single consensus vector that updates the current point. We propose several improvements: (i) a simple new method for calculating the consensus vector, (ii) a predictor-corrector approach to adjusting the consensus vector, (iii) an improved way of selecting the output point, and (iv) ways of selecting subsets of the constraints to operate on at a given iteration. These techniques greatly improve the performance of barrier method local solvers. Quadratic feasibility vectors are also investigated. Empirical results are given for a large set of nonlinear and nonconvex models.

Spatial joint constraints for the absolute nodal coordinate formulation using the non-generalized intermediate coordinates

In this investigation, a systematic procedure that can be used for modeling joint constraints for the absolute nodal coordinate formulation is developed. To this end, the non-generalized intermediate coordinates are introduced to derive a mapping between the generalized gradient coordinates and the non-generalized rotation parameters. With this mapping, a wide variety of joint constraints can be defined for the absolute nodal coordinate formulation in terms of the non-generalized reference coordinates and, therefore, existing well-developed constraint libraries formulated for the rigid body reference coordinates can be directly employed without significant modifications in existing codes. Furthermore, in order to define a rigid surface at the joint definition point, a set of orthonormality conditions is imposed on the gradient coordinates. This leads to not only accurate modeling of interface to mechanical joint, but also a simpler definition of the joint coordinate system obtained by the orthonormal gradient vectors. For this reason, a simpler form of constraint Jacobian and quadratic velocity vectors can be obtained as compared to those of the existing approach which requires the use of highly nonlinear joint coordinate system. A systematic procedure for eliminating the non-generalized coordinates and the dependent Lagrange multipliers associated with the coordinate mapping equations from the equations of motion is presented. As a result, a standard augmented form of the equations of motion can be obtained in terms of the generalized coordinates only. Several numerical examples are presented in order to demonstrate the use of the joint constraint formulation developed in this investigation.

붐(Boom)의 탄성을 고려한 해상크레인의 비선형 정적/동적 거동을 위한 수치 해석

해상크레인은 크레인을 탑재한 선박으로서, 조선소에서 대형 블록이나 구조물의 탑재 및 해상 운송 작업에 사용된다. 본 논문에서는 해상크레인과 중량물의 전후 동요(Surge), 상하 동요(Heave), 종 동요(Pitch)에 대한 정적/동적 거동을 분석하였다. 이 때, 유연 다물체계 동역학을 적용하여 해상크레인의 붐(boom)을 탄성으로 고려하였으며, 플로팅 프레임(floating frame)과 노드 좌표(nodal coordinates)를 사용하였다. 질량 행렬, 탄성 강성 행렬, 2 차 속도 벡터, 일반화 좌표 방향으로 작용하는 외력 등을 고려하여 모든 운동이 연성된 비선형 운동 방정식을 구성하였다. 외력으로는 비선형 유체정역학 힘, 선형화된 유체동역학 힘, wire rope 힘, 계류력이 고려되었다. 수치 해석을 위해 Hilber-Hughes-Taylor 방법을 비선형 운동방정식에 적용하였다. 정적 거동 분석을 통한 정적 평형 자세를 고려한 경우와 고려하지 않은 경우에 대해 결과를 비교하였으며, 수치 해석 방법에 대한 정적/동적 거동 분석 결과를 비교하였다.

On the algebraic structure of quadratic and bilinear dynamical systems

A number of important control and estimation problems in the field of aerospace and avionics involve dynamical models that are quadratic and bilinear functions of system states and inputs. In this paper, formal definitions of free and forced quadratic/bilinear dynamical systems are given and the algebraic structure of this class of systems is explored with a view to setting up a systematic approach for deriving state and measurement models. Properties of quadratic and bilinear vectors are investigated, and relationships between these and linear vectors established. Systematic procedure for constructing quadratic state and bilinear state–input vectors is derived. The quadratic/bilinear vector modeling technique is applied to the formulation of a state–space model for a second-order approximation of a general non-linear system.

Twisted modules for vertex operator algebras and Bernoulli polynomials

This work is a continuation of a series of papers of two of the present authors [L3], [L4], [M1]–[M3], stimulated by work of Bloch [Bl]. In those papers we used the general theory of vertex operator algebras to study central extensions of classical Lie algebras and superalgebras of differential operators on the circle in connection with values of ζ–functions at the negative integers. In the present paper, using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call “modified weak associativity.” This paper is an announcement. The detailed proofs will appear elsewhere. More specifically, the present goal is to obtain a new general Jacobi identity for twisted operators, and for related iterates of such operators, extending the previous analogous results in the untwisted setting in our papers mentioned above. As a consequence we obtain twisted constructions of certain central extensions of Lie algebras of differential operators on the circle, combining and extending methods from [L3], [L4], [M1]–[M3], [FLM1], [FLM2] and [DL]. In those earlier papers we used vertex operator techniques to analyze untwisted actions of the Lie algebra D, studied in [Bl], on a module for a Heisenberg Lie algebra of a certain standard type, based on a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form. Now consider an arbitrary isometry ν of period say p, that is, with ν = 1. Here we announce that the corresponding ν–twisted modules carry an action of the Lie algebra D in terms of twisted vertex operators, parametrized by certain quadratic vectors in the untwisted module. In particular, we extend a result from [FLM1], [FLM2], [DL] where actions of the Virasoro algebra were constructed using twisted vertex operators. In addition we explicitly compute certain “correction” terms for the generators of the “Cartan subalgebra” of D that naturally appear in any twisted construction. These correction terms are expressed in terms of special values of certain Bernoulli polynomials. They can in principle be generated, in the theory of vertex operator algebras, by the formal operator ex [FLM1], B.D. gratefully acknowledges partial support from an NSERC Postgraduate Scholarship. J.L. and A.M. gratefully acknowledge partial support from NSF grant DMS-0070800.

Principles of Multivariate Analysis, A User's Perspective.

Part I: Looking at multivariate data Part II: Samples, populations, and models Part III: Analysing ungrouped data Part IV: Analysing grouped data Part V: Analysing association among variables Appendix: some basic matrix theory A1 Definitions A2 Elementary arithmetic operations A3 Determinants and inverses A4 Quadratic forms A5 Latent roots and vectors A6 Matrix square root A7 Partitioned matrices A8 Vector differentiation References Index

A Classification of Quadratic Vectors

A Classification of Quadratic Vectors