Expression For Vector Research Articles

The Generalized Sidelobe Canceller Based on Quaternion Widely Linear Processing

We investigate the problem of quaternion beamforming based on widely linear processing. First, a quaternion model of linear symmetric array with two-component electromagnetic (EM) vector sensors is presented. Based on array's quaternion model, we propose the general expression of a quaternion semiwidely linear (QSWL) beamformer. Unlike the complex widely linear beamformer, the QSWL beamformer is based on the simultaneous operation on the quaternion vector, which is composed of two jointly proper complex vectors, and its involution counterpart. Second, we propose a useful implementation of QSWL beamformer, that is, QSWL generalized sidelobe canceller (GSC), and derive the simple expressions of the weight vectors. The QSWL GSC consists of two-stage beamformers. By designing the weight vectors of two-stage beamformers, the interference is completely canceled in the output of QSWL GSC and the desired signal is not distorted. We derive the array's gain expression and analyze the performance of the QSWL GSC in the presence of one type of interference. The advantage of QSWL GSC is that the main beam can always point to the desired signal's direction and the robustness to DOA mismatch is improved. Finally, simulations are used to verify the performance of the proposed QSWL GSC.

On the angular velocity of a rigid body: Matrix and vector representations

Abstract The angular velocity is an important property associated with the velocity state of moving rigid bodies. Unlike the velocity vector of a point, angular velocity vector is not in general equal to the time derivative of any single vector. Hence a unified, simple and comprehensible treatment of the subject may benefit the velocity analysis of complex multibody systems. This paper contributes with a new point of view of matrix and vector representations of angular velocity from the very foundations of classical kinematics of rigid bodies. This contribution was given a systematic, integrated and unified treatment, thus allowing the derivations to be based upon quantities which are expressed in terms of geometric objects (vectors) and geometric operations (vector addition, dot, and cross product). As a result, the approach leads naturally to simple and particularly useful expressions for the angular velocity vector, which allow a readily extension to three important representations involving the position and velocity of three noncollinear points pertaining to a moving rigid body.

Optical Nonresonance Vector Solitons in Dispersive and Kerr Media

A theory of optical nonresonance vector pulsing solitons in a Kerr media is considered. By using the multiple scale perturbative reduction method the wave equation is transformed to the coupled nonlinear Schrodinger equations. The shape of the optical nonresonance vector pulsing soliton with the difference and sum of the frequencies is presented. Explicit analytical expressions for the optical two-component vector pulsing soliton with phase modulation are obtained. It is shown that the two-component pulsing soliton in this special case can be transformed to the scalar pulsing soliton, and these waves have different shapes.

A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures

The purpose of the paper is to present a corotational beam element for the nonlinear dynamic analysis of 3D flexible frames. The novelty of the formulation lies in the use of the corotational framework (i.e., the decomposition into rigid body motion and pure deformation) to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix. As a consequence, cubic interpolations are adopted to formulate both inertia and internal local terms. In the derivation of the dynamic terms, an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained. To enhance the efficiency of the iterative procedure, an approximate expression of the tangent dynamic matrix is adopted. Four numerical examples are considered to assess the performance of the new formulation against the one suggested by Simo and Vu-Quoc (1988) [48]. It was observed that the proposed formulation proves to combine accuracy with efficiency. In particular, the present approach achieves the same level of accuracy as the formulation of Simo and Vu-Quoc but with a significantly smaller number of elements.

Third-order aberration analysis of tilted-pupil optical systems

The aberration field of an optical system with a tilted pupil is explored through expanding the vector expressions of the third-order wavefront aberrations. First, the vector forms of the wavefront aberrations are modified to obtain the aberration expressions with the pupil tilted; full field displays of coma and astigmatism in this situation are given. Then, the third-order aberration formulas with the pupil decentered and tilted simultaneously are derived and discussed. Finally, an example is taken to certify the validity of aberration distribution properties.

Multiple scattering of arbitrarily incident Bessel beams by random discrete particles

In this paper, we introduce an efficient numerical method to characterize the multiple scattering by random discrete particles illuminated by Bessel beams with arbitrary incidence. Specifically, the vector expressions of Bessel beams that perfectly satisfy Maxwell's equations in combination with rotation Euler angles are used to represent the arbitrarily incident Bessel beams. A hybrid vector finite element-boundary integral-characteristic-basis function method is utilized to formulate the scattering problems involving multiple discrete particles with a random distribution. Due to the flexibility of the finite element method, the adopted method can conveniently deal with the problems of multiple scattering by randomly distributed homogeneous particles, inhomogeneous particles, and anisotropic particles. Some numerical results are included to illustrate the validity and capability of the proposed method and to show the scattering behaviors of random discrete particles when they are illuminated by Bessel beams.

New Vector Description of Kinetic Pressures on Shaft Bearings of a Rigid Body Nonlinear Dynamics with Coupled Rotations around No Intersecting Axes

New vector description of kinetic pressures on shaft bearings of a rigid body nonlinear dynamics with coupled rotations around no intersecting axes is first main result presented in this paper. Mass moment vectors and vector rotators coupled for pole and oriented axes, defined by K. Hedrih in 1991, are used for obtaining vector expressions for kinetic pressures on the shaft bearings of a rigid body dynamics with coupled rotations around no intersecting axes. A complete analysis of obtained vector expressions for kinetic pressures on shaft bearings give us a series of the kinematical vectors rotators around both directions determined by axes of the rigid body coupled rotations around no intersecting axes. As an example of defined dynamics, we take into consideration a heavy gyro-rotor- disk with one degree of freedom and coupled rotations when one component of rotation is programmed by constant angular velocity. For this system with nonlinear dynamics, series of graphical presentation transformations in realizations with changes of eccentricity and angle of inclination (skew position) of heavy rigid body-disk in relation to self rotation axis are presented, as well as in realization with changing orthogonal distance between axes of coupled rotations. Angular velocity of kinetic pressures components in vector form are expressed by using angular acceleration and angular velocity of component coupled rotations of gyrorotor-disk.

Torsion Axial-Vector and Gravitational Energy for a Stiff Fluid Model in the Theory of Teleparallel Gravity

In this paper, we find the teleparallel version of the cylindrically symmetric stiff fluid space-time. The expressions for torsion vector and torsion axial-vector are obtained. We show that the value of torsion axial-vector depends on the choice of tetrad fields. Furthermore, we calculate the energy and momentum densities and show that these values do not depend on the choice of tetrad fields. Finally, we find the equation determining trajectory and spin precession of a Dirac equation in the space-time under consideration and show that the corresponding Hamiltonian depends on the choice of tetrad fields.

Local interfacial velocity measurement method using a four-sensor probe

This paper presents a theoretical foundation of the measurement methods for the instantaneous local interfacial velocity vector and the time-averaged local interfacial area concentration using a four-sensor probe for multi-dimensional two-phase flow measurements. The measurement method is derived based on a large bubble assumption that locally views the front and rear interfaces of an approaching bubble as two tangent planes. The newly-derived method provides an explicit expression for the instantaneous local interfacial velocity vector using a four-sensor probe. The derived method for the time-averaged local interfacial area concentration was found to be in the same form as that proposed by Kataoka et al. (1986) [1]. The derived method was applied to the practical two-phase flow measurements in a vertical pipe with an inner diameter of 200mm. The measured void fraction and interfacial velocity component in the axial direction were checked against the void fraction measurement using differential pressure gages and the superficial gas velocity measurement using gas flow meters, respectively. The measured interfacial velocity components in the radial and circumferential velocity components were found to be close to zero, which is in accordance with the fact that no stable flow of bubbles with certain horizontal velocity component exists in a vertical circular pipe. The measured interfacial area concentrations showed reasonable radial distributions in the pipe. The good agreements in the practical measurements suggest that the newly-derived method can reasonably measure interfacial velocity vector and interfacial area concentration in multi-dimensional two-phase flows.

VEST: Abstract vector calculus simplification in Mathematica

We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and vector expressions of a very general type to a well defined standard form. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by reduction, subsequently applying these to simplify large expressions. In a companion paper Burby et al. (2013) [12], we employ VEST in the automation of the calculation of high-order Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations. Program summaryProgram title: VEST (Vector Einstein Summation Tools)Catalogue identifier: AEQN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEQN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 10469No. of bytes in distributed program, including test data, etc.: 72539Distribution format: tar.gzProgramming language: Mathematica.Computer: Any computer running Mathematica.Operating system: Linux, Unix, Windows, Mac OS X.RAM: Usually under 10 MbytesClassification: 5, 12, 19.Nature of problem:Large scale vector calculus computationsSolution method:Reduce expressions to standard form in index notation, automatic derivation of multi-term vector identities.Restrictions:Current version cannot derive vector identities without cross products or curlAdditional comments:Intuitive user input and output in a combination of vector and index notationRunning time:Reduction to standard form is usually less than one second. Simplification of very large expressions can take much longer.

Scattering of a zero-order Bessel beam by arbitrarily shaped homogeneous dielectric particles

In this paper, we introduce an efficient numerical method based on surface integral equations to characterize the scattering of a zero-order Bessel beam by arbitrarily shaped homogeneous dielectric particles. The incident beam is described by vector expressions in terms of the electric and magnetic fields that perfectly satisfy Maxwell's equations. The scattering problems involving homogeneous dielectric particles with arbitrary shapes are formulated with the electric and magnetic current combined-field integral equation and modeled by using surface triangular patches. Solutions are performed iteratively by using the multilevel fast multipole algorithm. Some numerical results are included to illustrate the validity and capability of the proposed method. These results are also expected to provide useful insights into the scattering of a Bessel beam by complex-shaped particles.

Deslauriers–Dubuc interpolating wavelet beam finite element

This paper presents the formulation of beam finite elements based on Deslauriers–Dubuc interpolating wavelets, also known as Interpolets. Unlike other wavelet families like Daubechies, Interpolets possess rational filter coefficients, are smooth, symmetric and therefore more suitable for use in numerical methods. Displacement and rotation shape functions are obtained and presented graphically. Expressions for stiffness matrix and force vector are developed based on connection coefficients, which are inner products of basis functions and their derivatives. In order to validate the formulation, several examples with increasing level of complexity are tested and results are compared with analytical and standard beam element solutions.

VECTOR METHOD BASED ON MASS MOMENT VECTORS AND VECTOR ROTATORS APPLIED TO RIGID-BODY MULTI-COUPLED ROTATIONS AROUND NONINTERSECTING AXES

The first part of the paper contains a short review of a series of published papers in the area of system dynamics with coupled rotations as well as of a series of author's various published research results in the area of vector method based on the mass inertia moment vectors and corresponding deviational vector components and vector rotators for the pole and oriented axis, introduced and defined by K. Hedrih in 1991. The vector with principal importance is vector of the rigid body mass inertia moment at the point and for the axis oriented by the unit vector, and with a corresponding component vector of the rigid body mass deviational moment for the rotation axis through the same pole. Second part presents the vector method based on mass moment vectors and vector rotators coupled for pole and oriented nonintersecting axes in application for obtaining vector expressions for kinetic parameters of a rigid body dynamics with multi-coupled rotations around nonintersecting axes. A complete analysis of obtained vector expressions for linear momentum and angular momentum and their corresponding derivatives gives us a series of the kinematical vector rotators around the corresponding directions determined by axes of the rigid body coupled multi-rotations around nonintersecting axes. Series of the theorems are defined.

Computing optimal base-stock levels for an inventory system with imperfect supply

We study a single-item, single-site, periodic-review inventory system with negligible fixed ordering costs. The supplier to this system is not entirely reliable, such that each order is a Bernoulli trial, meaning that, with a given probability, the supplier delivers the current order and any accumulated backorders at the end of the current period, resulting in a Geometric distribution for the actual resupply lead time. We develop a recursive expression for the steady-state probability vector of a discrete-time Markov process (DTMP) model of this imperfect-supply inventory system. We use this recursive expression to prove the convexity of the inventory system objective function, and also to prove the optimality of our computational procedure for finding the optimal base-stock level. We present a service-constrained version of the problem and show how the computation of the optimal base-stock level using our DTMP method, incorporating the explicit distribution of demand over the lead time plus review (LTR) period, compares to approaches in the literature that approximate this distribution. We also show that the version of the problem employing an explicit penalty cost can be solved in closed-form for the optimal base-stock level for two specific period demand distributions, and we explore the behavior of the optimal base-stock level and the corresponding optimal service level under various values of the problem parameters.

Exact Analytical Expression for Outgoing Intensity from the Top of the Atmosphere in a Flat Homogeneous Atmosphere–Ocean System using Analytical Discrete Ordinate method for the Solution of Polarized Radiation Transfer Equation.

This research is a part of the work devoted on the application of analytical Discrete Ordinate (ADO) method to the polarized monochromatic radiative transfer equation undergoing anisotropic scattering with source function matrix in a finite coupled Atmosphere -Ocean media having flat interface boundary conditions involving specular reflection and transmission matrix. Discontinuities in the derivatives of the Stokes vector with respect to the cosine of the polar angle at smooth interface between the two media with different refractive indices (air and water) is tackled by using a suitable quadrature scheme devised earlier. Atmosphere and ocean are assumed to be homogeneous. No stratification is adopted in the two media. Exact expression for the emergent radiation intensity vector from the top of the atmosphere is derived. Exact expressions for the emergent polarized radiation intensity vector from the air-water interface as well as from any point of the two medium in any direction can also be derived in terms of eigenvectors and eigenvalues. Keywords: Polarized Radiative transfer, Eigenvectors, Eigenvalues, Green function, specular.

Bounds for discrete tomography solutions

We consider the reconstruction of a function on a finite subset of Z2 where the line sums in certain directions are prescribed. Its real solutions form a linear manifold, its integer solutions a grid. First we provide an explicit expression for the projection vector from the origin onto the linear solution manifold in the case of only row and column sums of a finite subset of Z2. Next we present a method for estimating the maximal distance between two binary solutions. Subsequently we deduce an upper bound for the distance from any given real solution to the nearest integer solution. This enables us to estimate the stability of solutions. Finally we generalize the first result mentioned above to the continuous case.

Orthogonal kernel projecting plane for radar HRRP recognition

It is known that the subspace method is effective for radar target recognition. Its key step is to find a suitable low-dimensional subspace. In this paper, a novel subspace method, namely orthogonal kernel projecting plane (OKPP), is proposed for radar target recognition using high-resolution range profile (HRRP). The goal of OKPP is to maximize the between-class distance while minimizing the within-class distance. By introducing an orthogonality constraint into the objective function, we obtain the orthogonal basis vectors of OKPP. Comparing with the conventional kernel-based subspace methods, such as kernel principal component analysis (KPCA) and kernel Fisher discriminant analysis (KFDA), the nonlinear features extracted by OKPP reduce redundancy and improve the target recognition performance. The explicit expressions of the basis vectors of OKPP can be solved without using singular value decomposition (SVD) process, and thus reduces the computation complexity. The experimental results using measured data show that the proposed method has an encouraging performance.

Surface polaritons with arbitrary magnetic and dielectric materials: new regimes, effects of negative index, and superconductors

A surface magnon-polariton can be excited by both p- and s-polarized light if at least one of the layers is a magnetic material. We present general expressions of the tangential wave vectors of s- and p-polarized light at an interface of two media. Analysis reveals additional new regimes of surface polariton resonances with magnetic materials for s- and p-polarized light. The tangential wave vectors are found to be equal in magnitude to the normal wave vectors at surface polariton resonances. The spatial distributions of the fields at resonant enhancement and the spectra of the tangential wave vectors are studied for different dielectric permittivities and magnetic permeabilities of the two media. If one of the media has dispersive dielectric function and permeability function, additional surface polariton resonance peaks appear for both s- and p polarizations. For a medium with a superconductor, the tangential component increases asymptotically at lower frequencies, providing subwavelength capability at the terahertz regime.

Improved infrared target-tracking algorithm based on mean shift

An improved IR target-tracking algorithm based on mean shift is proposed herein, which combines the mean-shift-based gradient-matched searching strategy with a feature-classification-based tracking algorithm. An improved target representation model is constructed by considering the likelihood ratio of the gray-level features of the target and local background as a weighted value of the original kernel histogram of the target region. An expression for the mean-shift vector in this model is derived, and a criterion for updating the model is presented. Experimental results show that the algorithm improves the shift weight of the target pixel gray level and suppresses background disturbance.

TORSION VECTOR AND TORSION AXIAL-VECTOR IN VAN STOCKUM SPACETIME

In this paper, we work with teleparallel gravity and show that the values of the torsion vector and the torsion axial-vector depend on the choice of the tetrad field. We choose two different sets of tetrad fields, each of them satisfies the tetrad's conditions and leads to the Van Stockum spacetime. We obtain expressions for the torsion vector and the torsion axial-vector. In addition, we found that the obtained expressions are quite different for the two sets of tetrad fields. For the spacetime under consideration, if the metric coefficients are assumed to be functions of t only, it is then shown that the torsion axial-vector vanishes completely and the torsion vector still depends on the choice of tetrad fields. Finally, we found the spin precession of a Dirac particle in the spacetime under consideration.