Projector Matrix Research Articles

Cartesian constraints in QM/MM optimizations.

With the rise of quantum mechanical/molecular mechanical (QM/MM) methods, the interest in the calculation of molecular assemblies has increased considerably. The structures and dynamics of such assemblies are usually governed to a large extend by intermolecular interactions. As a result, the corresponding potential energy surfaces are topological rich and possess many shallow minima. Therefore, local structure optimizations of QM/MM molecular assemblies can be challenging, in particular if optimization constraints are imposed. To overcome this problem, structure optimization in normal coordinate space is advocated. To do so, the external degrees of freedom of a molecule are separated from the internal ones by a projector matrix in the space of the Cartesian coordinates. Here we extend this approach to Cartesian constraints. To this end, we devise an algorithm that adds the Cartesian constraints directly to the projector matrix and in this way eliminates them from the reduced coordinate space in which the molecule is optimized. To analyze the performance and stability of the constrained optimization algorithm in normal coordinate space, we present constrained minimizations of small molecular systems and amino acids in gas phase as well as water employing QM/MM constrained optimizations. All calculations are performed in the framework of auxiliary density functional theory as implemented in the program deMon2k.

Projector matrix product operators, anyons and higher relative commutants of subfactors

A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck–Mariën–Williamson–Şahinoğlu-Haegeman–Verstraete on two-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of length k is naturally identified with the kth higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between two-dimensional topological order and subfactor theory.

Про підхід до визначення відповідності між пікселями камери і проєктора у мультимедійному тирі

Предметом вивчення в статті є геометричні перетворення у системі«проєктор–екран–камера» для визначення відповідності між пікселямикамери і проєктора у мультимедійному тирі. Метою є розробка математичної моделі та алгоритму визначення відповідності між пікселямикамери і проєктора. Це потрібно для співставлення положення центроїдалазерної плями від пострілу в матриці камери і мішені, що генеруєтьсяу матриці проєктора. Завдання: формалізувати задачу виникнення геометричних спотворень у системі «проєктор–екран–камера»; розробитиматематичну модель визначення відповідності між пікселями камериі проєктора; вибрати ефективний алгоритм її вирішення. Використовуваними методами є: математична модель вирівнювання зображення наоснові гомографії, метод бінаризації з вибором значення порога методом Оцу; метод Дугласа–Пекера, який зменшує кількість точок, що апроксимують криву. Отримано такі результати. Згідно аналізу геометричних спотворень у системі «проєктор–екран–камера» сформульовано задачу розробки моделі для вирівнювання зображення з метоювизначення відповідності між пікселями камери і проєктора. Розроблено математичну модель і алгоритм вирівнювання точок зображення мішені у матриці камери з точками зображення мішені у матриці проєктора. Розроблено і програмно реалізовано алгоритм визначення відповідності між пікселями камери і проєктора. Математичнізалежності для визначення відповідності між пікселями камери і проєктора встановлено на основі матриці гомографії. Коефіцієнти матриціобчислюються за відповідними кутовими точками прямокутника матриці проєктора і його спотвореного трапецеїдального зображення наматриці камери. Розроблено алгоритм автоматичного визначення вершин трапеції та встановлення відповідності між пікселями камериі проєктора. Проведено аналіз та експериментальні дослідження факторів, що впливають на точність алгоритму: точність визначення вершин трапеції, за якими знаходяться коефіцієнти матриці гомографії; ступіньзаповнення області огляду камери зображенням із проєктора; відповідність роздільної здатності камери і проєктора. Дано рекомендації щодозменшення їхнього впливу. Зроблено такі висновки. Наукова новизнаотриманих результатів полягає в наступному: розроблено і експериментально досліджено математичну модель визначення відповідності міжпікселями камери і проєктора у мультимедійному тирі через вирівнюваннязображення, що відображається з екрана проєктора у камері, на основігомографії. До цього для вирівнювання зображень у мультимедійному тирі використовувалась модель на основі 2D-перетворень, у якій взаємнеположення у просторі проєктора, екрана і камери не враховується.Оскільки у гомографії враховуються тільки лінійні перетворення, у подальшому планується удосконалити модель — врахувати нелінійніспотворення, що виникають у об’єктивах камери і проєктора.

On the Dynamics of Matrix Models for Immune Clonal Networks

We suggest some new matrix models for clonal network dynamics which are typical for simulating many biological clonal type networks and study their dynamics to the stable states. We describe in detail and derives the corresponding matrix equations governing immune system dynamics based on the general gradient type principles that can be inherent to a wide class of real living objects. Clonal networks of a special type are modelled by symmetric projector matrix variables simultaneously taking into account both asymmetry of the interaction to each other and adaptation states that can be realized owing to possible idiotypic clonal suppressions. We perform computer simulations of the model dynamics for some simple cases of relatively low dimension, paying special attention to the dynamics of amounts of activated receptor strings within a clonal network.

Two projector triple products in quantum crystallography

AbstractConsider a projector matrixP, representing the first order reduced density matrixin a basis of orthonormal atom‐centric basis functions. A mathematical question arises, and that is, how to breakPinto its natural component kernel projector matrices, while preservingN‐representability of. The answer relies upon 2‐projector triple products,P′jPP′j. The triple product solutions, applicable within the quantum crystallography of large molecules, are determined by a new form of the Clinton equations, which—in their original form—have long been used to ensureN‐representability of density matrices consistent with X‐ray diffraction scattering factors. As such, the goal of this paper is to outline a possible pathway for the application of quantum crystallography to crystals of large molecular systems.

On the Generalization of the Random Projection Method for Problems of the Recovery of Object Signal Described by Models of Convolution Type

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.

Monte Carlo methods for estimating Mallows's Cp and AIC criteria for PLSR models. Illustration on agronomic spectroscopic NIR data

AbstractMallows's Cp and Akaike information criterion (AIC) are common criteria for selecting the dimensionality of regression models, as an alternative to cross‐validation (CV) and nonparametric bootstrap. A key parameter in the calculation of Cp and AIC is the effective number of degrees of freedom of the model, or model complexity (d). Parameter d is generally easy to calculate for linear smoothers, that is, models for which the prediction of the training response y is given by = S y where S is a projector matrix that does not involve y. Nevertheless, d is more difficult to estimate for nonlinear smoothers, such as partial least squares regression (PLSR). In this article, we present two algorithms for estimating d for PLSR based on Monte Carlo simulation methods (parametric bootstrap and perturbation analysis) and with the particular case of high dimensional data. We compare these Monte Carlo methods to three other algorithms already published. We used the d estimates to compute Cp and AIC and select PLSR model dimensionalities that we then compare to CV. Two real and heterogeneous agronomic near infrared (NIR) datasets were considered as examples.

The direct force correction based framework for general co-rotational analysis

The use of nonlinear projector matrix in co-rotational (CR) analysis was pioneered by Rankin and Nour-Omid in 1990s (Rankin and Nour-Omid, 1988; Nour-Omid and Rankin, 1991), and has almost became a standard manner for CR formulations deduction over the past thirty years. This matrix however relies heavily on a hysterical and sophisticated derivation of the variation of the local displacements to the global ones, leading to complicated expressions for the internal force vector and the tangent stiffness matrix, which may devalue the simplicity and convenience for the original intention of using CR approach. This paper begins by making a discussion on existing element independent CR formulation and the objective is to develop a new and simpler framework for general CR analysis that avoids using conventional nonlinear projector matrix. The methodology consists of two steps in the element internal force calculation. The first one is to obtain a preliminary result of the internal force. This is done by following the conventional element-independent CR formulation but dropping the terms involving projector matrix and therefore yields simple formulations of the internal force and the tangent stiffness matrix. The second one is a correction step to obtain a new internal force vector that satisfies the element self-equilibrium condition. This step inherits the spirit of using projector matrix but is conducted directly in the global frame, thus avoiding complicated entanglement of local–global rotation and is independent of the choice of the local CR frame used in the CR analysis. This further leads to a simple and unified formulation for different kinds of elements that can be cooperated in CR framework. Closed formulation of the correction force as well as the related consistent tangent stiffness matrix is derived for different correction approaches. It is also shown that the existing linear projector matrix used for infinitesimal rotation analysis is a special case of the current correction approaches. Multiple numerical examples involving various kinds of elements and different choices of element local CR frame are presented to demonstrate the performance of the proposed framework. The outcomes show that for all the examples the accuracy of the results is comparable with those obtained in conjunction with conventional nonlinear projector matrix.

Numerical Quantification of Controllability in the Null Space for Redundant Manipulators

Redundant motion, which is possible when robotic manipulators are over-actuated, can be used to control robot arms for a wide range of tasks. One of the best known methods for controlling redundancy is the null space projection, which assigns a priority while executing desired tasks. However, when the manipulator is projected into null space, its motion would be limited, since the motion is only permitted in the direction that does not interfere with the primary task. In this study, we have analyzed the null space projector matrix to derive the appropriate direction of the redundant motion by quantifying the allowed motion in each direction. As a result, we have found an ellipsoidal boundary, in which the redundant motion is permitted to move. We have named this ellipsoidal boundary as ’null space quality’ in directions. The proposed null space quality shows similar aspects with that of the robot manipulability, but it reveals a decisively different value when the manipulator operates within the null space. The experimental results showed that the robotic manipulator tracked the sinusoidal input trajectory with reduced root mean square (RMS) error by 33.84%. Furthermore, we have demonstrated the obstacle avoidance of a robotic arm utilizing the null space projector while considering the null space quality.

Строго косые проекторы и их свойства

In this paper, strictly oblique projectors are defined as projectors that cannot be represented as the sum of two projectors, one of which is a nonzero orthoprojector. A theorem is proved that each projector can be represented in a unique way as the sum of a strictly oblique projector and an orthoprojector. The properties of such projectors are given. For example: if the projector is strictly oblique, then its Hermitian adjoint is also strictly oblique; the rank of a strictly oblique projector is at most n/2, where n is the order of the projector matrix; the property of the projector to be strictly oblique is preserved with a unitary similarity. The work is a continuation of the previous work of the authors, the main result of which is such a matrix expression for an arbitrary projector: where A and B are two matrices of full rank whose columns define range and the null space of this projector. Based on this result, the article shows that the strictly oblique part of any projector P is given by the expression: P(P – P+P)+P. And equality P = P(P – P+P)+P is a criterion that the projector P is a strictly oblique projector. The decomposition of the projector obtained in the work is applied to the practical problem of oblique projection onto the plane

Fractional Charlier moments for image reconstruction and image watermarking

In this paper, we propose a new set of discrete orthogonal polynomials called fractional Charlier polynomials (FrCPs). This new set will be used as a basic function to define the fractional discrete orthogonal Charlier moments (FrCMs). The proposed FrCPs are derived algebraically using the spectral decomposition of Charlier polynomials (CPs), then the Lagrange interpolation formula is used to derive the spectral projectors. Then, each spectral projector matrix is decomposed by the singular value decomposition (SVD) technique in order to build a basic set of orthonormal eigenvectors which help to develop FrCPs. FrCMs are deduced in matrix form from the proposed FrCPs and are applied for image reconstruction and watermarking. The experimental results show the capacity of the FrCMs proposed for image reconstruction and image watermarking against different attacks such as noise and geometric distortions.

Tight frames, Hadamard matrices and Zauner’s conjecture

We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and complex Hadamard matrix in dimension d2, and a pair of ETFs (equiangular tight frames) in dimensions . We also show that a WH (Weyl–Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d. We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds are non-linear. Restricted defect calculations are consistent with this being true for the structures associated to every known SIC with d between 3 and 16, suggesting it may be true for all .

Robustness of subspace-based algorithms with respect to the distribution of the noise: Application to DOA estimation

This paper addresses the theoretical analysis of the robustness of subspace-based algorithms with respect to non-Gaussian noise distributions using perturbation expansions. Its purpose is twofold. It aims, first, to derive the asymptotic distribution of the estimated projector matrix obtained from the sample covariance matrix (SCM) for arbitrary distributions of the useful signal and the noise. It proves that this distribution depends only of the second-order statistics of the useful signal, but also on the second and fourth-order statistics of the noise. Second, it derives the asymptotic distribution of the estimated projector matrix obtained from any M-estimate of the covariance matrix for both real (RES) and complex elliptical symmetric (CES) distributed observations. Applied to the MUSIC algorithm for direction-of-arrival (DOA) estimation, these theoretical results allow us to theoretically evaluate the performance loss of this algorithm for heavy-tailed noise distributions when it is based on the SCM, which is significant for weak signal-to-noise ratio (SNR) or closely spaced sources. These results also make it possible to prove that this performance loss can be alleviated by replacing the SCM by an M-estimate of the covariance for CES distributed observations, which has been observed until now only by numerical experiments.

An efficient solution procedure for constraint-free elastic structure subject to self-equilibrium static loading

The stiffiness matrix would be singular in finite element static analysis of free structure, so the analysis would fail. To solve this problem, we can obtain the total displacement of free structure under some well-posed constraint, and then use the projector matrix constructed by rigid body modes (RBM) to separate the elastic displacement from the total displacement. Another point, the traditional RBM matrix is only linear, which isn’t satisfy the large rotational condition, so a new construction of RBM matrix based on rigid body motion is proposed. It can be found that the nonlinear projector matrix can be reduced to linear projector matrix in some cases. The numerical results show the feasibility of the proposed method and the differences in linear and nonlinear analysis.

AN EFFICIENT CO-ROTATIONAL FEM FORMULATION USING A PROJECTOR MATRIX

Co-rotational finite element (FE) formulations can be seen as a very efficient approach to resolving geometrically nonlinear problems in the field of structural mechanics. A number of co-rotational FE formulations have been well documented for shell and beam structures in the available literature. The purpose of this paper is to present a co-rotational FEM formulation for fast and highly efficient computation of large three-dimensional elastic deformations. On the one hand, the approach aims at a simple way of separating the element rigid-body rotation and the elastic deformational part by means of the polar decomposition of deformation gradient. On the other hand, a consistent linearization is introduced to derive the internal force vector and the tangent stiffness matrix based on the total Lagrangian formulation. It results in a non-linear projector matrix. In this way, it ensures the force equilibrium of each element and enables a relatively straightforward upgrade of the finite elements for linear analysis to the finite elements for geometrically non-linear analysis. In this work, a simple 4-node tetrahedral element is used. To demonstrate the efficiency and accuracy of the proposed formulation, nonlinear results from ABAQUS are used as a reference.

Stability of switched positive descriptor systems with average dwell time switching

In this paper, the problems of stability for a class of switched positive descriptor systems (SPDSs) with average dwell time (ADT) switching are investigated. First, based on the equivalent switched system and the properties of the projector matrix, sufficient stabilities are given for the underlying systems in both continuous-time and discrete-time contexts. Then, a sufficient stability condition for the SPDS with both stable and unstable subsystems is obtained. The stability results for the SPDSs are represented in terms of a set of linear programmings (LPs) by the multiple linear co-positive Lyapunov function (MLCLF) approach. Finally, three numerical examples are given to illustrate the effectiveness of the obtained theoretical results.

Spatial light modulator based on liquid-crystal video projector matrix for information processing systems

Experimental results of the optical properties investigation of the liquid crystal video projector matrix are given as well as its application for the input of images into the coherent optical Fourier processor and into the joint Fourier transform correlator. It is shown that such modulators are perspective for the application in the coherent optical processing systems

Game theory approach to state and input simultaneous estimation for discrete-time LTV systems

This article aims to provide a simple approach realising state observation and input estimation simultaneously for discrete-time LTV systems in H∞ setting. Through solving a two-player zero sum differential game, appealing results are obtained in two folds. First, necessary and sufficient solvability conditions for state and input simultaneous estimation problem are given in terms of solution to a set of difference Riccati recursion. Second, one estimator is presented with special innovation structure, where innovation information is used to update state observation tuned by gain matrix and to provide input estimation through a projector matrix, where gain matrix and projector matrix are constructed from solution to difference Riccati recursion. At last, simulation results are provided to justify proposed approach.

Reverberation nulling and echo enhancement at low frequency using waveguide invariance

Based on the fact that the transfer function vector between a source receiver array and the dominant scatterer of boundary reverberation at a range can be obtained from the corresponding reverberations scattered from this range cell, a reverberation nulling concept using time reversal processing has been proposed. However, current reverberation nulling methods have certain limitations when applied into practice, which would null boundary reverberation and target echo simultaneously. As a solution, a passive reverberation nulling and echo enhancement method at low frequency using waveguide invariance is proposed in this paper. In this method, the reverberation subspace for the target range cell is not obtained directly from the return signals scattered by the target range cell but from the return signals scattered by a range cell located before the target using waveguide invariance, so as to suppress the reverberation embodied in the target echo by passive reverberation nulling. Besides, a range-dependent optimal weighting vector rather than conventional projector matrix is deduced to null the reverberation component meanwhile maximizing the target echo, thereby enhancing the echo-to-reverberation ratio furthest. Numerical simulations in typical range-independent shallow water environment demonstrate the efficacy and the improved performance of the proposed method for echo-to-reverberation enhancement.

Managing logical channel configurations in complex satellite architectures

AbstractAs a result of the increased switching complexity of many communications satellites, often transponders are available with a variety of configuration options offering flexibility in power, bandwidth, and coverage. Many of these configurations share resources that prevent them from being simultaneously active, making it a challenge to understand and manage the satellite capacity inventory. In this paper we define the logical channel configuration (LCC) as an unambiguous unit of the inventory and present a novel method of modelling the interdependencies between these LCCs using linear algebra. In particular, a projector matrix is defined which can be used to update the activation state of all LCCs when the state of one or more of these is changed. As a result, a number of valid LCC scenarios can be easily generated and evaluated for optimal use of the satellite resources. Copyright © 2005 John Wiley & Sons, Ltd.