Abstract: We study decompositions of functions in the Hardy spaces into linear combinations of the basic functions in the orthogonal rational systems Bn, which are obtained in the respective contexts through Gram–Schmidt orthogonalization process on shifted Cauchy kernels. Those lead to adaptive decompositions of quaternionic-valued signals of finite energy. This study is a generalization of the main results of the first author's recent research in relation to adaptive Takenaka–Malmquist systems in one complex variable. Copyright © 2011 John Wiley & Sons, Ltd.

Abstract: A method introduced by Mayer (Theor Chem Acc 104:163, 2000) for generating an orthogonal set of basis vectors, perpendicular to an arbitrary start vector, is examined. The procedure provides the complementary vectors in closed form, expressed with the components of the start vector. Mayer’s method belongs to the family of orthogonalization schemes, which keep an arbitrary vector intact without introducing any non-physical sequence-dependence. It is shown that Mayer’s orthogonalization is recovered by performing a two-step combination of the Gram-Schmidt and Lowdin’s symmetrical orthogonalization. Processor time requirement of constructing Mayer’s orthonormal set is proportional to ∼N2, in contrast to the rough ∼N3 CPU requirement of performing either a full Gram-Schmidt or Lowdin’s symmetrical orthogonalization. Utility of Mayer’s orthogonalization is demonstrated on an electronic structure application using perturbation theory to improve multiconfigurational wavefunctions.

Abstract: A new family of hierarchical divergence-conforming vector bases is proposed for tetrahedral, prism and brick cells. These functions span the divergence-conforming reduced-curl spaces of Nedelec. The bases are constructed from orthogonal scalar polynomials to enhance their linear independence, which is a simpler process than an orthogonalization applied to the final vector functions. Specific functions are tabulated to order 6.5. These basis are intended for use with numerical solutions of volume integral equations or differential equations containing divergence operators.

Abstract: In this correspondence, a method is presented for estimating double-selective channels using superimposed training (ST). The estimator is based on a subspace projection of the time-varying channel onto a set of two dimensional orthogonal functions. These functions are formed via the outer product of the discrete prolate spheroidal basis vectors and the universal basis vectors. This approach allows the channel to be expanded in both the time-delay and time dimensions with the fewest parameters when incomplete channel statistics are given. This correspondence also provides a theoretical performance analysis of the estimation algorithm and its corroboration via simulations. It is shown that this new method provides an enhancement in channel estimation when compared with state-of-the-art approaches.

Abstract: The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries---like midplane symmetry---are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane, and (under certain circumstances) the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as, for instance, the method of Teng and Edwards. In a preceding paper, it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces. Unfortunately, the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all conceivable cases. Hence, a systematic derivation of a more general treatment seemed advisable. In a second paper, the author suggested the use of real Dirac matrices as basic tools for coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues. In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. The decoupling of symplectic matrices which are exponentials of such Hamiltonian matrices can be deduced from this in a few steps. It is shown that this algebraic decoupling is closely related to a geometric ``decoupling'' by the orthogonalization of the vectors $\stackrel{\ensuremath{\rightarrow}}{E}$, $\stackrel{\ensuremath{\rightarrow}}{B}$, and $\stackrel{\ensuremath{\rightarrow}}{P}$, which were introduced with the so-called ``electromechanical equivalence.'' A mathematical analysis of the problem can be traced down to the task of finding a structure-preserving block diagonalization of symplectic or Hamiltonian matrices. Structure preservation means in this context that the (sequence of) transformations must be symplectic and hence canonical. When used iteratively, the decoupling algorithm can also be applied to $n$-dimensional systems and requires $\mathcal{O}({n}^{2})$ iterations to converge to a given precision.

Abstract: This paper reflects results of research related to developing a new methodology for automatic graph drawing based on applying genetic algorithms The methodology has permitted the elaboration of a hybrid technique that combines the most popular, classical, topology-shape-metric approach to orthogonal drawings on the grid and a genetic algorithm that is directed, in its evolutionary process, at multicriteria decision making in a fuzzy environment In the traditional use of the topology-shape-metric approach, a single fixed planar embedding is obtained in the planarization step Thereafter this embedding is subjected to the orthogonalization and compaction steps However, this sequence does not guarantee that the fixed planar embedding will generate a final drawing of a good quality Moreover, every topology-shape-metric step is classified as a NP-hard problem, and choices as well as heuristics used in previous stages have a direct impact on subsequent ones Taking this into account, the developed hybrid technique generates a greater number of planar embeddings by varying the order of edges' insertion when forming the planar embeddings in planarization step Thus, the problem is formulated as a permutation-based combinatorial optimization problem The genetic algorithm is applied at the planarization step of the topology-shape-metric This allows one to generate the population with the corresponding number of planar embeddings Each planar embedding obtained in the planarization step is submitted to the orthogonalization and compaction Their results serve for applying the procedures of multicriteria decision making in a fuzzy environment Thus, in the evolutionary process, the genetic algorithm is able to select individuals, which provide more harmonious solutions (relatively of the solutions obtained with traditional applying the topology-shape-metric approach) from the point of view of the aesthetic criteria that are usually utilized at the three steps of automatic graph drawing This is convincingly demonstrated by experimental results given in the paper

Abstract: Independent component analysis (ICA), instead of the traditional discrete cosine transform (DCT), is often used to project log Mel spectrum in robust speech feature extraction. The paper proposed using symmetric orthogonalization in ICA for projecting log Mel spectrum into a new feature space as a substitute in extracting speech features to solve the problem of cumulative error and unequal weights that deflation orthogonalization brings, so as to improve the robustness of speech recognition systems, and increase the efficiency of estimation at the same time. Furthermore, the paper studied the nonlinearities of the objective function in ICA and their coefficients, tested them in all kinds of environments , finding that they influenced the recognition rate greatly in speech recognition systems, and applied a new coefficient in the proposed method. Experiments based on HMM and Aurora-2 speech corpus suggested that the new method was superior to deflation-based ICA and MFCC.

Abstract: This paper proposes an orthogonalization of the Hilbert matrix in element matrices of the infinite edge elements. The validity of the infinite edge element is demonstrated in previous researches, but the Hilbert matrix results in extremely slow convergence in the ICCG method, especially when using higher order expansions. The proposed orthogonalization technique improves the convergence drastically and it makes the infinite elements practical in the electromagnetic FEM analysis of the open boundary problems in quasi-static magnetic fields.

Abstract: Two transformations are proposed that give orthogonal components with a one-to-one correspondence between the original vectors and the components. The aim is that each component should be close to the vector with which it is paired, orthogonality imposing a constraint. The transformations lead to a variety of new statistical methods, including a unified approach to the identification and diagnosis of collinearities, a method of setting prior weights for Bayesian model averaging, and a means of calculating an upper bound for a multivariate Chebychev inequality. One transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors, and is determined using a new algorithm that also provides the decomposition of a positive-definite matrix in terms of a diagonal matrix and a correlation matrix. The algorithm is shown to converge to a global optimum.

Abstract: A method in a mobile communication terminal (24) includes receiving a Multi-User Multiple-Input Multiple-Output (MU-MIMO) signal, which includes at least a precoded transmission that is addressed to the terminal and which includes one or more Demodulation Reference Signals (DM-RS). Control information, which indicates one or more scrambling sequences and one or more orthogonalization sequences used in producing the DM-RS, is received in the mobile communication terminal. The control information is provided in a format that does not permit signaling of every possible scrambling sequence and orthogonalization sequence selection. The control information is interpreted in the mobile communication terminal so as to identify the scrambling sequences and orthogonalization sequences used in producing the DM-RS, and the DM-RS are demodulated using the identified scrambling sequences and orthogonalization sequences.

Abstract: By employing the regularized block diagonalization (RBD) preprocessing technique, the multi-user multi-input multi-output (MU-MIMO) broadcast channel is decomposed into multiple parallel independent single user multi-input multi-output (SU-MIMO) channels and achieves the maximum diversity order at high data rates. The computational complexity of RBD, however, is relatively high due to two singular value decomposition (SVD) operations. In this paper, a low-complexity lattice reduction aided RBD is proposed. The first SVD is replaced by a QR decomposition, and the orthogonalization procedure provided by the second SVD is substituted by a lattice reduction whose complexity is mainly contributed by a QR decomposition. Simulation results show that the proposed algorithm can achieve almost the same sum-rate as RBD while offering a lower complexity and substantial BER gains with perfect as well as imperfect channel state information at the transmit side.

Abstract: The invention provides a data preprocessing based covariance matrix orthogonalization wave-beam forming method aiming at solving the problem that the conventional covariance matrix based GS (Gram-Schmidt) orthogonalization (RGS) algorithm can not be directly used for training snapshot and contains desired signal information and belonging to the technical field of adaptive wave-beam forming The data preprocessing based covariance matrix orthogonalization wave-beam forming method comprises the following steps of: firstly preprocessing training snapshot, and rejecting a desired signal; then estimating the covariance matrix by utilizing preprocessed data, and forming an interference subspace by carrying out GS orthogonalization on lines of the covariance matrix; and finally carrying out orthogonal projection on a corresponding static weight vector towards the interference subspace to obtain an adaptive weight vector In the invention, in order to more accurately estimate the interference subspace, an adaptive threshold of orthogonalization is corrected on the basis of preprocessing; and when the training snapshot is mixed with the desired signal, the data preprocessing based covariance matrix orthogonalization wave-beam forming method provided by the invention can greatly enhance the anti-interference property of an array

Abstract: The three-dimensional theory of elasticity is used for a study of the stress-strain state in a hollow cylinder with varying stiffness. The corresponding problem is solved by a method that is partly analytical and partly numerical in nature: Spline approximations and collocation are used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of higher order for the radial coordinate, which is then solved using the method of stable discrete orthogonalization. Results for an inhomogeneous cylinder for various types of stiffness are presented.

Abstract: We propose a novel distributed QR factorization algorithm for orthogonalizing a set of vectors in a wireless sensor network. The algorithm originates from the classical Gram-Schmidt orthogonalization which we formulate in a distributed way using the dynamic consensus algorithm. In contrast to existing distributed QR factorization algorithms, all elements of matrices Q and R are computed simultaneously and updated iteratively after each transmission. Assuming synchronous message broadcasting and communication only with neighboring nodes without any central computing unit (fusion center), we prove convergence of the algorithm. We investigate the algorithm in terms of numerical accuracy and we discuss the influence of the initial data distribution on the algorithm performance. Moreover, we provide a comparison with existing distributed QR algorithms in terms of communication cost and memory requirements, and we illustrate the comparison by simulations.

Abstract: A bounded linear operator T on Banach space is subspace-supercyclic for a nonzero subspace M if there exists a vector whose projective orbit intersects the subspace M in a relatively dense set.We constructed examples to show that subspace-supercyclic is not a strictly infinite dimensional phenomenon,and that some subspace-supercyclic operators are not supercyclic.We provided a subspace-supercyclicity criterion and offered two necessary and sufficient conditions for a path of bounded linear operators to have a dense G_δset of common subspace-hypercyclic vectors and common subspace-supercyclic vectors.

Abstract: This report is a sequel of the publications [1] [3] [2]. We consider the multiobjective optimization problem of the simultaneous minimization of n (n ≥ 2) criteria, {J_i(Y)}(i=1,...,n), assumed to be smooth real-valued functions of the design vector Y ∈ OMEGA ⊂ R^N (n ≤ N) where OMEGA is the (open) admissible domain of R^N over which these functions admit gradients. Given a design point Y^0 ∈ OMEGA that is not Pareto-stationary, we introduce the gradients {J_i'}(i=1,...,n) at Y = Y^0, and assume them to be linearly independent. We also consider the possible "scaling factors", {S_i} (i=1,...,n) (S_i > 0 , ∀i), as specified appropriate normalization constants for the gradients. Then we show that the Gram-Schmidt orthogonalization process, if conducted with a particular calibration of the normalization, yields a new set of orthogonal vectors {u_i} (i=1,..,n) spanning the same subspace as the original gradients; additionally, the minimum-norm element of the convex hull corresponding to this new family, omega, is calculated explicitly, and the Frechet derivatives of the criteria in the direction of omega are all equal and positive. This direct process simplifies the implementation of the previously-defined Multiple-Gradient Descent Algorithm (MGDA).

Abstract: This report is a sequel to several publications in which a Multiple-Gradient Descent Algorithm (MGDA), has been proposed and tested for the treatment of multi-objective differentiable optimization. Originally introduced in [2], the method has been tested and reformulated in [6]. Its efficacy to identify the Pareto front has been demonstrated in [7], in comparison with an evolutionary strategy. Recently, a variant, MGDA-II, has been proposed in which the descent direction is calculated by a direct procedure [4] based on a Gram-Schmidt orthogonalization process (GSP) with special normalization. This algorithm was tested in the context of a simulation by domain partitioning, as a technique to match the different interface components concurrently [3]. The experimentation revealed the importance of scaling, and a slightly modified normalization procedure was proposed ("MGDA-IIb"). In this new report, two novel variants are proposed. The first, MGDA-III, realizes two enhancements. Firstly, the GSP is conducted incompletely whenever a test reveals that the current estimate of the direction of search is adequate also w.r.t. the gradients not yet taken into account; this improvement simplifies the identification of the search direction when the gradients point roughly in the same direction, and makes the Frechet derivative common to several objective-functions larger. Secondly, the order in which the different gradients are considered in the GSP is defined in a unique way devised to favor an incomplete GSP. In the second variant, MGDA-IV, the question of scaling is addressed when the Hessians are known. A variant is also proposed in which the Hessians are estimated by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula.

Abstract: As a result of mutual coupling, the embedded element patterns of non-regular arrays are all different. It is known that, with the help of Macro Basis Functions, array patterns can be written as a finite series of pattern multiplication problems. However, for the very large arrays envisaged in SKA AA-lo, this still leads to a prohibitive number of calibration coefficients. Two elements may help in this respect: (i) the setting of main lobe and first few sidelobes as a priority for calibration and (ii) the orthogonalization of patterns of MBFs. Examples are shown for arrays of wideband dipoles and of bowtie antennas. The issue of relative shift-independance of the different array factors upon scaning is also analyzed.

Abstract: Bounded Component Analysis (BCA) has recently been introduced as an alternative linear decomposition scheme. In this approach the boundedness property of sources is exploited to replace the usual independence assumption with a weaker assumption, which enables development of methods to separate both independent and dependent components from their mixtures. This paper positions total output range minimization based blind source separation approach as a BCA method for the separation of uncorrelated sources. It is shown that the global minimizers of the corresponding optimization problem are the perfect separators. Furthermore, a stationary point analysis for the corresponding algorithms based on symmetrical orthogonalization is provided. The main result of this analysis is that the range minimization based parallel BCA algorithm and the kurtosis maximization based Independent Component Analysis algorithm have related set of identified stationary points.

Abstract: When the desired signal is mixed in the training data, the conventional Gram-Schmidt orthogonalization of covariance matrix (RGS) adaptive beamforming will result in the desired signal cancellation. Therefore, a modified Gram-Schmidt orthogonalization of covariance matrix (MRGS) adaptive beamforming based on data preprocessing is proposed in this paper. In the proposed algorithm, the training data are firstly preprocessed to remove the desired signal, in the following the corresponding covariance matrix is estimated, and the interference subspace is reconstructed by using the Gram-Schmidt orthogonalization of the columns of modified covariance matrix. Finally, the adaptive weight vector is obtained by orthogonally projecting the quiescent weight vector into the interference subspace. Moreover, the adaptive threshold of the preprocessed data is modified correspondingly for more accurate interference subspace estimation. According to the simulations, it is found that the proposed MRGS adaptive beamforming algorithm can improve the performance significantly.

Abstract: Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping theorems. These spectral mapping theorems are inclusive in general. Equality can be established in the self-adjoint case. The arising functions are shown to possess a biradial character. It is shown that to any given set of Jacobi parameters corresponds a biradial measure yielding these parameters in an iterative orthogonalization process in this function space, once equipped with the corresponding $L^2$ structure.