Diagonal Decomposition Research Articles

Spectral properties of a functional binomial matrix

In this article, we consider the definition of the Fibonacci polynomial sequence with the second-order linear recurrence relation, where coefficients and initial conditions depend on the variable $t$. And then, we introduce the functional binomial matrix depending on the coefficients of the second-order linear recurrence relation. In the following, we study the spectral properties of the functional binomial matrix using the Fibonacci polynomial sequence and we obtain a diagonal decomposition for it using the Vandermunde matrix. Finally, by applying some linear algebra tools we obtain a number of combinatorial identities involving the Fibonacci polynomial sequence.

Diagonal decompositions of shapes and their algebras

AbstractThe formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, andm-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.

Some spectral properties of the non-backtracking matrix of a graph

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. In doing so, we create a block diagonal decomposition of the non-backtracking matrix, more clearly expressing its eigenvalues. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. Specifically, an upper-bound on the spectral radius is found in terms of the number of nodes and edges of the graph. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.

High‐dimensional covariance matrix estimation using a low‐rank and diagonal decomposition

AbstractWe study high‐dimensional covariance/precision matrix estimation under the assumption that the covariance/precision matrix can be decomposed into a low‐rank component and a diagonal component . The rank of can either be chosen to be small or controlled by a penalty function. Under moderate conditions on the population covariance/precision matrix itself and on the penalty function, we prove some consistency results for our estimators. A block‐wise coordinate descent algorithm, which iteratively updates and , is then proposed to obtain the estimator in practice. Finally, various numerical experiments are presented; using simulated data, we show that our estimator performs quite well in terms of the Kullback–Leibler loss; using stock return data, we show that our method can be applied to obtain enhanced solutions to the Markowitz portfolio selection problem. The Canadian Journal of Statistics 48: 308–337; 2020 © 2019 Statistical Society of Canada

DIADECOMP: A new approach to analyze decompositions from projection spectroscopy

We demonstrate for the first time a complete small protein characterization with the projection-decomposition approach, including full assignments as well as determination of the 3D fold. In TOCSY- and NOESY-type 4D experiments, pairing of signals from hydrogens and from their respective heavy atoms in decompositions represents a new problem. An approach, referred to as “DIADECOMP” (diagonal decomposition), is introduced to solve this problem; it consists of two separate decompositions of the input projections, differing in a 45° rotation of the spectral axes. While DIADECOMP requires a somewhat complex formulation, in practice it results in observing signals in the rotated decompositions that correspond to sums or differences of frequencies. When applied to a small protein, human defensin β6, the analysis of a HCC(CO)NH-TOCSY with DIADECOMP results in largely unambiguous assignments of the aliphatic side chain groups. Furthermore, DIADECOMP applied to a 15N-HSQC-NOESY-15N-HSQC provides all expected short distances between amide groups (defined as all HN-HN distances <3.5Å in a reference structure). It is worth noting that short HN-HN distances unambiguously define α-helices, the alignment of β-strands in sheets, as well as the presence of β-bulges. This approach of using a minimal amount of NMR data, namely four projection experiments recorded in ∼2.5days, resulted for the human defensin β6 in complete assignments and a backbone fold with a RMSD of the non-flexible structure of 0.6Å. Uniqueness of decompositions specifically from TOCSY- and NOESY-type 4D experiments is discussed.

Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints that arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a “dual” method to the convexification method employing the perspective function on semicontinuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of mixed-integer quadratic programming solvers when applied to the perspective reformulations of the problem.

A simple block diagonal precoding for multi-user MIMO broadcast channels

Abstract The block diagonalization (BD) is a linear precoding technique for multi-user multi-input multi-output (MIMO) broadcast channels, which is able to completely eliminate the multi-user interference (MUI), but it is not computationally efficient. In this paper, we propose the block diagonal Jacket matrix decomposition, which is able not only to extend the conventional block diagonal channel decomposition but also to achieve the MIMO broadcast channel capacity. We also prove that the QR algorithm achieves the same sum rate as that of the conventional BD scheme. The complexity analysis shows that our proposal is more efficient than the conventional BD method in terms of the number of the required computation.

Max-Max Based Internal Model Control Method for Multivariable System with Time Delay

Multivariable system with time delay and coupling widely exist in industrial which may destroy the normal work of control system. An unconventional internal model controller design method will be introduced in this paper. The closed loop system can be decouple by calculate the inverse of transfer function matrix and the optimal diagonal decomposition matrix. Finally, this method was applied in a multivariable system with different time delays, the simulation results can show the effectiveness of this method.

Interpretation of magnetic data in the Sinop area of Mid Black Sea, Turkey, using tilt derivative, Euler deconvolution, and discrete wavelet transform

article i nfo Article history: Received 8 March 2011 Accepted 25 May 2011 Available online xxxx The identification of lineaments in magnetic anomaly data is an important step in interpretation. In this paper, three methods for locating magnetic lineaments are applied to total field magnetic anomaly data of the Sinop area of mid Black Sea, northern Turkey. Based upon the variation in magnetic lineaments, structural map of the study area was obtained. The tilt derivative (TDR), Euler deconvolution (ED) and discrete wavelet transform (DWT) have been of great utility in the interpretation of magnetic anomaly data. These three methods were applied to enhance magnetic anomalies due to anomalous sources in part of the basement complex of study area. The TDR obtained from the magnetic anomaly reduced to pole (RTP) was designed to map geologic contacts. The source boundaries and depths are determined from the zero contours, and the half distance between ± π/4 contours or the distance between zero and + π/4 or − π/4 contour of TDR, respectively. The ED technique is used to estimate the source depth at the semi-infinite contact location. The DWT technique was carried out to infer the substructure orientation. The direction of maxima and minima coefficients of the horizontal, vertical and diagonal decompositions of the DWT works in imaging the horizontal locations of the anomalous sources. The results allow us to image and characterize subsurface lineaments for the survey area.

Subspace-Based Noise Reduction for Speech Signals via Diagonal and Triangular Matrix Decompositions: Survey and Analysis

We survey the definitions and use of rank-revealing matrix decompositions in single-channel noise reduction algorithms for speech signals. Our algorithms are based on the rank-reduction paradigm and, in particular, signal subspace techniques. The focus is on practical working algorithms, using both diagonal (eigenvalue and singular value) decompositions and rank-revealing triangular decompositions (ULV, URV, VSV, ULLV, and ULLIV). In addition, we show how the subspace-based algorithms can be analyzed and compared by means of simple FIR filter interpretations. The algorithms are illustrated with working Matlab code and applications in speech processing.

Analysis of new pivoting strategy for the LDLT decomposition on a multiprocessor system with distributed memory

It is well known that optimal control techniques can provide the ability to design suitable strategies, however, the on-line computing requirements are excessive. The normal procedure is to make various assumptions so that the processing demands are reduced. Based on these assumptions, sequences of linear-quadratic-performance optimal control problems need to be considered. These in turn give rise to standard two-point-boundary-value problems. The solution to such problems involves the computation of the algebraic Riccati equations (AREs). The block diagonal decomposition LDLT, is the key step for those algorithms based on the matrix sign function that are used in solving the AREs. The last few years have witnessed a tremendous effort towards the development of reliable algorithms to solve the AREs and apply it in industrial situations. However, all the implementations and testing of the proposed algorithms have been performed on powerful machines thereby limiting their practical application. The authors present a pivoting strategy that: (i) requires only one-dimension of the matrix for the selection of the pivot; (ii) generates regular communication patterns; and (iii) establishes a software mechanism for the development of fault tolerant applications. The results obtained from a multiprocessor system with a one-way ring topology indicate that the block diagonal LDLT decomposition is a true candidate for real-time use and fault tolerant applications and also as a framework-test for the LAPACK library.

An Evaluation of Four Different Wavelet Decomposition Procedures for Spatial Feature Discrimination in Urban Areas

Texture or spatial arrangement of neighborhood objects and features plays an important role in the human visual system for pattern recognition and image classification. The traditional spectral–based image processing techniques have proven inadequate for urban land use and land cover mapping from images acquired by the current generation of fine–resolution satellites. This is because of the high frequency spatial arrangements or complex nature of urban features. There is a need for an effective algorithm to digitally classify urban land use and land cover categories using high–resolution image data. Recent studies using wavelet transforms for texture analysis have generally reported better accuracy. Based on a high–resolution ATLAS image, this study illustrates four different wavelet decomposition procedures – the standard, horizontal, vertical, and diagonal decompositions – for urban land use and land cover feature extraction with the use of 33×33 pixel samples. The standard decomposition approach was found to be the most efficient approach in urban texture analysis and classification. For comparison purposes and to better evaluate the accuracy of wavelet approaches in image classification, spatial autocorrelation techniques (Moran’s I and Geary’s C) and the spatial co–occurrence matrix method were also examined. The results suggest that the wavelet transform approach is superior to all other approaches.

Generalized Hardy operators and normalizing measures

.to be bounded from Lo[0 o) to Lq(s) are given. Here a(s) and b(s) are similarly ordered functions and k(s,y) satisfies a modified GHO condition. Nearly block diagonal decompositions of positive operators are introduced as is the concept of a normalizing measure. An application is made to estimates for the remainder in a Taylor approximation.

Parallel solutions of very large sparse Lyapunov equations by balanced BBD decompositions

In this paper, a method for the parallel solution of large sparse matrix equations is presented. The method is based on the balanced bordered block diagonal (BBD) decomposition which is applied in conjunction with the successive over-relaxation (SOR) iterative method to solve large Lyapunov equations in the Kronecker sum representation. A variety of experimental results will be presented, including solutions of Lyapunov equations with matrices as large as 1993/spl times/1993.

On Linear Least-Squares Problems with Diagonally Dominant Weight Matrices

The solution of the unconstrained weighted linear least-squares problem is known to be a convex combination of the basic solutions formed by the nonsingular subsystems if the weight matrix is diagonal and positive definite. In particular, this implies that the norm of this solution is uniformly bounded for any diagonal and positive definite weight matrix. In addition, the solution set is known to be the relative interior of a finite set of polytopes if the weight matrix varies over the set of positive definite diagonal matrices. In this paper, these results are reviewed and generalized to the set of weight matrices that are symmetric, positive semidefinite, and diagonally dominant and that give unique solution to the least-squares problem. This is done by means of a particular symmetric diagonal decomposition of the weight matrix, giving a finite number of diagonally weighted problems but in a space of higher dimension. Extensions to equality-constrained weighted linear least-squares problems are given. A discussion of why the boundedness properties do not hold for general symmetric positive definite weight matrices is given. The motivation for this research is from interior methods for optimization.

Parallel Iterative Solution of Very Large Sparse Lyapunov Equations

In this paper, we present a method for the parallel solution of large sparse Lyapunov equations. The method is based on the balanced Bordered Block Diagonal (BBD) decomposition, which is known to produce matrix structures that are ideally suited for parallel computation. Experimental results will be presented for a Lyapunov equation with matrices of dimension 1,993 × 1,993.

A nested decomposition algorithm for parallel computations of very large sparse systems

In this paper we present a generalization of the balanced border block diagonal (BBD) decomposition algorithm, which was developed for the parallel computation of sparse systems of linear equations. The efficiency of the new procedure is substantially higher, and it extends the applicability of the BBD decomposition to extremely large problems. Examples of the decomposition are provided for matrices as large as 250, 000 × 250, 000, and its performance is compared to other sparse decompositions. Applications to the parallel solution of sparse systems are discussed for a variety of engineering problems.

Quantum-mechanical computers and uncomputability.

The time evolution operator for any quantum-mechanical computer is diagonalizable, but to obtain the diagonal decomposition of a program state of the computer is as hard as actually performing the computation corresponding to the program. In particular, if a quantum-mechanical system is capable of universal computation, then the diagonal decomposition of program states is uncomputable. As a result, in a universe in which local variables support universal computation, a quantum-mechanical theory for that universe that supplies its spectrum cannot supply the spectral decomposition of the computational variables. A ``theory of everything'' can be simultaneously correct and fundamentally incomplete.

CONFORMALLY REDUCED WZNW THEORY, NEW EXTENDED CHIRAL ALGEBRAS AND THEIR ASSOCIATED TODA TYPE INTEGRABLE SYSTEMS

We propose and analyze a large class of conformal reductions Cons [g(H, d)] of WZNW theory based on the integral gradations of the underlying Lie algebra g. The W bases of the associated W algebras W[g(H, d)] are constructed under the generalized Drinfeld-Sokolov gauge which we call O’Raifeartaigh gauge of the constrained Kac-Moody currents, and the equations of motion of the extended Toda type integrable systems corresponding to these W algebras are also derived. As an example, we construct explicitly the W algebra associated with the (pqp) block diagonal decomposition of sl2p+q, namely W[(pqp)2], and discuss some of the properties thereof.

Finite symmetric systems and their analysis

Symmetry of linear mechanical systems permits one to substantially reduce the effort in their analysis. Classification and structural analysis of the finite linear symmetric systems (models) are studied in this paper. There are two variants of the symmetry approach: mechanical and algebraic. In accordance with the former, a symmetric system is replaced by one or several small nonsymmetric subsystems which are subjected to special loads obtained from the initial set. The total response of the original symmetric system is found by special superposition of partial responses of these subsystems. The algebraic approach is based on the explicit block diagonal decomposition of the matrix equation corresponding to a symmetric system. While both approaches have the same efficiency the latter is easier to implement and describe. It is presented here. The condtions under which the symmetry technique may be utilized do not include symmetry of the applied loads (specifically, symmetry of those loads which form the right side of the associated equations). Nevertheless, if the loads are symmetric, the efficiency of the symmetry approach substantially increases. Group theory, which is widely used in this paper, is the mathematical tool for the study of symmetry, and all necessary notions are introduced.