Subspace Of Product Research Articles

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.

Deriving the cone fundamentals: a subspace intersection method.

Two ideas, proposed by Thomas Young and James Clerk Maxwell, form the foundations of colour science: (i) three types of retinal receptors encode light under daytime conditions, and (ii) colour matching experiments establish the critical spectral properties of this encoding. Experimental quantification of these ideas is used in international colour standards. However, for many years, the field did not reach consensus on the spectral properties of the biological substrate of colour matching: the spectral sensitivity of the cone fundamentals. By combining auxiliary data (thresholds, inert pigment analyses), complex calculations, and colour matching from genetically analysed dichromats, the human cone fundamentals have now been standardized. Here, we describe a new computational method to estimate the cone fundamentals using only colour matching from the three types of dichromatic observers. We show that it is not necessary to include data from trichromatic observers in the analysis or to know the primary lights used in the matching experiments. Remarkably, it is even possible to estimate the fundamentals by combining data from experiments using different, unknown primaries. We then suggest how the new method may be applied to colour management in modern image systems.

Subarray-based joint source localization in shallow water waveguide via subspace intersection

By the use of an underwater bottom-mounted Horizontal Linear Array (HLA), the two-dimensional joint localization technique and its accuracy are analyzed in this article. The proposed method takes advantage of arrival angles measured by a dual-subarray system consisting of a large-aperture HLA. Then, for the system, expressions for source position estimate and error variation are derived. Analysis of the Geometrical Dilution of Precision, which is a measure of localization error defined in Eq. (4), reveals that the localization error is mainly related to the source bearing, subarray interval and their measurement errors. In order to investigate the impact of the underwater waveguide, numerical simulations were carried out based on the normal mode theory. By using the distribution of acoustic intensity and structure of modal arrival, it is proved that the localization accuracy increases as the grazing angle and the subarray interval increase. However, when the measurement errors of source bearing and subarray interval increase, the localization accuracy will decrease. In addition, the obtained bearing estimation result provides an evident characteristic of beam spreading. When sources are near the end-fire direction of the dual-subarray system, significant bearing estimation error occurs, the negative effect of which is then presented. By applying the Subspace Intersection method, unbiased bearing estimation has been achieved, with relatively reduced computational complexity. Most of all, the improved localization performance is validated by experimental data collected in the South China Sea.

A data-driven fault detection approach for unknown large-scale systems based on GA-SVM

This paper is concerned with the data-driven fault detection (FD) problem for large-scale systems with unknown system matrices and interconnection signals. Due to the existence of the unknown system matrices and interconnection terms, the FD problem of large-scale systems is hardly solved by the existing model-based methods. To tackle this problem, the interconnection terms are firstly estimated by the subspace intersection technique. Then, based on the estimated interconnection terms and input-output data, the support vector machine (SVM) is constructed to design fault detector for each subsystem. Furthermore, genetic algorithm (GA) is introduced to optimize the parameters of SVM such that FD performance is improved. Finally, a numerical example is adopted to illustrate the effectiveness of the developed FD approach.

Sequential Design-Space Reduction and Its Application to Hull-Form Optimization

Hull-form optimization is a complex engineering problem. Owing to the several numerical simulations and complex design-performance spaces, hull-form optimization is considered an inefficient process, which makes determining the global optimum difficult. This study used rough set theory (RST) to acquire knowledge and reduce the design space for hull-form optimization. Furthermore, we studied one of the hull-form optimization problems by practically applying RST to the appropriate number of sampling points. To solve this problem, we proposed the RST-based sequential design-space reduction (SDSR) method that uses interval theory to calculate subspace intersections and unions, as well as test calculations to choose an appropriate stopping criterion. Finally, SDSR was used to optimize a KRISO container ship to minimize the wave-making resistance. The results were compared to those of direct optimization and one-time design-space reduction, thus proving the feasibility of this method.

Continuous limits of generalized pentagram maps

We provide a rigorous treatment of continuous limits for various generalizations of the pentagram map on polygons in RPd by means of quantum calculus. Describing this limit in detail for the case of the short-diagonal pentagram map, we verify that this construction yields the (2,d+1)-KdV equation, and moreover, the Lax form of the pentagram map in the limit is proved to become the Lax representation of the corresponding KdV system. More generally, we introduce the χ-pentagram map, a geometric construction defining curve evolutions by directly taking intersections of subspaces through specified points. We show that its different configurations yield certain other KdV equations and provide an argument towards disproving the conjecture that any KdV-type equation can be discretized through pentagram-type maps.

Generalized Canonical Correlation Analysis: A Subspace Intersection Approach

Generalized Canonical Correlation Analysis (GCCA) is an important tool that finds numerous applications in data mining, machine learning, and artificial intelligence. It aims at finding `common' random variables that are strongly correlated across multiple feature representations (views) of the same set of entities. CCA and to a lesser extent GCCA have been studied from the statistical and algorithmic points of view, but not as much from the standpoint of linear algebra. This paper offers a fresh algebraic perspective of GCCA based on a (bi-)linear generative model that naturally captures its essence. It is shown that from a linear algebra point of view, GCCA is tantamount to subspace intersection; and conditions under which the common subspace of the different views is identifiable are provided. A novel GCCA algorithm is proposed based on subspace intersection, which scales up to handle large GCCA tasks. Synthetic as well as real data experiments are provided to showcase the effectiveness of the proposed approach.

Robust Adaptive Beamforming Based on Covariance Matrix Reconstruction With Annular Uncertainty Set and Vector Space Projection

The performance of adaptive beamforming will degrade dramatically in practical application due to system errors including signal direction error, array geometry error, gain, and phase errors. Besides, the performance will further deteriorate when the desired signal is contained in the sample data. Therefore, a robust adaptive beamforming method based on the covariance matrix reconstruction with annular uncertainty set (AUS) and vector space projection (VSP) is proposed in this letter. By integrating the corresponding Capon spectrum over the surface of AUS, the interference-plus-noise covariance matrix (INCM) and the desired signal covariance matrix are reconstructed, respectively. Because the steering vector (SV) of desired signal lies in the intersection of the signal subspaces of the sample covariance matrix and the reconstructed desired signal covariance matrix, it can be estimated by the VSP method. Finally, the adaptive weight vector is calculated based on the reconstructed INCM and the estimated SV. Simulation and experiment results show that the proposed method can effectively suppress the interference and achieve excellent performance under system errors.

Circumcentered Methods Induced by Isometries

Motivated by the circumcentered Douglas–Rachford method recently introduced by Behling, Bello Cruz and Santos to accelerate the Douglas–Rachford method, we study the properness of the circumcenter mapping and the circumcenter method induced by isometries. Applying the demiclosedness principle for circumcenter mappings, we present weak convergence results for circumcentered isometry methods, which include the Douglas– Rachford method (DRM) and circumcentered reflection methods as special instances. We provide sufficient conditions for the linear convergence of circumcentered isometry/reflection methods. We explore the convergence rate of circumcentered reflection methods by considering the required number of iterations and as well as run time as our performance measures. Performance profiles on circumcentered reflection methods, DRM and method of alternating projections for finding the best approximation to the intersection of linear subspaces are presented.

Identification of Physical Helicopter Models Using Subspace Identification

Subspace identification is a powerful tool due to its well-understood techniques based on linear algebra (orthogonal projections and intersections of subspaces) and numerical methods like singular value decomposition. However, the state space model matrices, which are obtained from conventional subspace identification algorithms, are not necessarily associated with the physical states. This can be an important deficiency when physical parameter estimation is essential. This holds for the area of helicopter flight dynamics, where physical parameter estimation is mainly conducted for mathematical model improvement, aerodynamic parameter validation, and flight controller tuning. The main objective of this study is to obtain helicopter physical parameters from subspace identification results. To achieve this objective, the subspace identification algorithm is implemented for a multirole combat helicopter using both FLIGHTLAB simulation and real flight-test data. After obtaining state space matrices via subspace identification, constrained nonlinear optimization methodologies are utilized for extracting the physical parameters. The state space matrices are transformed into equivalent physical forms via the "sequential quadratic programming" nonlinear optimization algorithm. The required objective function is generated by summing the square of similarity transformation equations. The constraints are selected with physical insight. Many runs are conducted for randomly selected initial conditions. It can be concluded that all of the significant parameters can be obtained with a high level of accuracy for the data obtained from the linear model. This strongly supports the idea behind this study. Results for the data obtained from the nonlinear model are also evaluated to be satisfactory in the light of statistical error analysis. Results for the real flight-test data are also evaluated to be good for the helicopter modes that are properly excited in the flight tests.

The block-wise circumcentered–reflection method

The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas--Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered-reflection method is able to both accelerate the average reflection scheme by the Douglas--Rachford method and cope with the intersection of more than two affine subspaces. We now introduce the technique of circumcentering in blocks, which, more than just an option over the basic algorithm of circumcenters, turns out to be an elegant manner of generalizing the method of alternating projections. Linear convergence for this novel block-wise circumcenter framework is derived and illustrated numerically. Furthermore, we prove that the original circumcentered-reflection method essentially finds the best approximation solution in one single step if the given affine subspaces are hyperplanes.

Robust Hyperbolic Chaos in Froude Pendulum with Delayed Feedback and Periodic Braking

We indicate a possibility of implementing hyperbolic chaos using a Froude pendulum that is able to produce self-oscillations due to the suspension on a shaft rotating at constant angular velocity, in the presence of time-delay feedback and of periodic braking by the application of additional frictional force. We formulate a mathematical model and carry out its numerical research. In the parameter space we reveal areas of chaotic and regular dynamics using the analysis of Lyapunov exponents and some other diagnostic tools. It is shown that there are regions in the parameter space where the Poincaré stroboscopic map has an attractor, which is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space. We confirm the hyperbolicity of the attractor by numerical calculations including the analysis of angles of intersections of stable and unstable invariant subspaces of vectors of small perturbations for trajectories on the attractor and verify the absence of tangencies between these subspaces.

Subspace identification of individual systems in a large-scale heterogeneous network

This paper considers the identification of a network consisting of discrete-time LTI systems that are interconnected by their unmeasurable states. For a large-scale network, the computational burden prevents a centralized solution. To cope with this problem, a subspace-based local identification approach using local observations is presented, which consists of subspace intersection operations in both the temporal and spatial domains. Sufficient conditions are provided for the consistent identification of the presented identification approach. Finally, the implementation of this approach on 1D network is specially investigated and numerical simulations are provided to show its effectiveness.

Formulas for calculating the dimensions of the sums and the intersections of a family of linear subspaces with applications

The union and the intersection of subspaces are fundamental operations in geometric algebra, while it is well known that both sum and intersection of linear subspaces in a vector space over a field are linear subspaces as well. A fundamental problem on a given family of linear subspaces is to determine the dimensions of their sum and intersection, as well as the orthogonal projectors onto the sum and the intersection of the family of linear subspaces. In this article, we first revisit an analytical formula for calculating the dimension of the intersection of a family of linear subspaces \({{\mathcal {M}}}_1, \, {{\mathcal {M}}}_2, \ldots , {\mathcal M}_k\) in a finite-dimensional vector space, and establish some analytical expressions of the orthogonal projector onto the intersection using generalized inverses of matrices. We then establish a general formula for calculating the dimension of the sum of the family of linear subspaces. We also give some applications of these formulas in determining the maximum and minimum dimensions of the intersections of column spaces of matrix products involving generalized inverses.

On relationships between two linear subspaces and two orthogonal projectors

AbstractSum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

Hyperbolic chaos in a system of two Froude pendulums with alternating periodic braking

We propose a new example of a system with a hyperbolic chaotic attractor. The system is composed of two coupled Froude pendulums placed on a common shaft rotating at constant angular velocity with braking by application of frictional force to one and other pendulum turn by turn periodically. A mathematical model is formulated and its numerical study is carried out. It is shown that attractor of the Poincaré stroboscopic map in a certain range of parameters is a Smale – Williams solenoid. The hyperbolicity of the attractor is confirmed by numerical calculations analyzing the angles of intersection of stable and unstable invariant subspaces of small perturbation vectors and verifying absence of tangencies between these subspaces.

Collaboration space division in collaborative product development based on a genetic algorithm

The advance in the global environment, rapidly changing markets, and information technology has created a new stage for design. In such an environment, one strategy for success is the Collaborative Product Development (CPD). Organizing people effectively is the goal of Collaborative Product Development, and it solves the problem with certain foreseeability. The development group activities are influenced not only by the methods and decisions available, but also by correlation among personnel. Grouping the personnel according to their correlation intensity is defined as collaboration space division (CSD). Upon establishment of a correlation matrix (CM) of personnel and an analysis of the collaboration space, the genetic algorithm (GA) and minimum description length (MDL) principle may be used as tools in optimizing collaboration space. The MDL principle is used in setting up an object function, and the GA is used as a methodology. The algorithm encodes spatial information as a chromosome in binary. After repetitious crossover, mutation, selection and multiplication, a robust chromosome is found, which can be decoded into an optimal collaboration space. This new method can calculate the members in sub-spaces and individual groupings within the staff. Furthermore, the intersection of sub-spaces and public persons belonging to all sub-spaces can be determined simultaneously.

Intersections of Maximal Subspaces of Zeros of Two Quadratic Forms

We calculate the dimensions of the intersections of maximal subspaces of zeros of a nonsingular pair of quadratic forms. We then count the number of sets of distinct such subspaces that intersect in a given dimension.

Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors

Starting from Anosov chaotic dynamics of geodesic flow on a surface of negative curvature, we develop and consider a number of self-oscillatory systems including those with hinged mechanical coupling of three rotators and a system of rotators interacting through a potential function. These results are used to design an electronic circuit for generation of rough (structurally stable) chaos. Results of numerical integration of the model equations of different degree of accuracy are presented and discussed. Also, circuit simulation of the electronic generator is provided using the NI Multisim environment. Portraits of attractors, waveforms of generated oscillations, Lyapunov exponents, and spectra are considered and found to be in good correspondence for the dynamics on the attractive sets of the self-oscillatory systems and for the original Anosov geodesic flow. The hyperbolic nature of the dynamics is tested numerically using a criterion based on statistics of angles of intersection of stable and unstable subspaces of the perturbation vectors at a reference phase trajectory on the attractor.

Geometric Arveson–Douglas Conjecture-Decomposition of Varieties

In this paper, we prove the Geometric Arveson–Douglas Conjecture for a special case that allows some singularity on \(\partial {\mathbb {B}_n}\). More precisely, we show that if a variety can be decomposed into two varieties, each having nice properties and intersecting nicely with \(\partial \mathbb {B}_n\), then the Geometric Arveson–Douglas Conjecture holds on this variety. We obtain this result by applying a result by Suarez, which allows us to “localize” the problem. Our result then follows from the simple case when the two varieties are intersection of linear subspaces with \(\mathbb {B}_n\).