Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 2002

# Search results for: eigenvalues of the covariance matrix

##### 2002 The Second Smallest Eigenvalue of Complete Tripartite Hypergraph

Abstract:

In the terminology of the hypergraph, there is a relation with the terminology graph. In the theory of graph, the edges connected two vertices. In otherwise, in hypergraph, the edges can connect more than two vertices. There is representation matrix of a graph such as adjacency matrix, Laplacian matrix, and incidence matrix. The adjacency matrix is symmetry matrix so that all eigenvalues is real. This matrix is a nonnegative matrix. The all diagonal entry from adjacency matrix is zero so that the trace is zero. Another representation matrix of the graph is the Laplacian matrix. Laplacian matrix is symmetry matrix and semidefinite positive so that all eigenvalues are real and non-negative. According to the spectral study in the graph, some that result is generalized to hypergraph. A hypergraph can be represented by a matrix such as adjacency, incidence, and Laplacian matrix. Throughout for this term, we use Laplacian matrix to represent a complete tripartite hypergraph. The aim from this research is to determine second smallest eigenvalues from this matrix and find a relation this eigenvalue with the connectivity of that hypergraph.

Keywords: connectivity, graph, hypergraph, Laplacian matrix

##### 2001 Parallel Computation of the Covariance-Matrix

Abstract:

We address the issues related to the computation of the covariance matrix. This matrix is likely to be ill conditioned following its canonical expression, thus consequently raises serious numerical issues. The underlying linear system, which therefore should be solved by means of iterative approaches, becomes computationally challenging. A huge number of iterations is expected in order to reach an acceptable level of convergence, necessary to meet the required accuracy of the computation. In addition, this linear system needs to be solved at each iteration following the general form of the covariance matrix. Putting all together, its comes that we need to compute as fast as possible the associated matrix-vector product. This is our purpose in the work, where we consider and discuss skillful formulations of the problem, then propose a parallel implementation of the matrix-vector product involved. Numerical and performance oriented discussions are provided based on experimental evaluations. Downloads 220
##### 2000 Bounds on the Laplacian Vertex PI Energy

Authors: Ezgi Kaya, A. Dilek Maden

Abstract:

A topological index is a number related to graph which is invariant under graph isomorphism. In theoretical chemistry, molecular structure descriptors (also called topological indices) are used for modeling physicochemical, pharmacologic, toxicologic, biological and other properties of chemical compounds. Let G be a graph with n vertices and m edges. For a given edge uv, the quantity nu(e) denotes the number of vertices closer to u than v, the quantity nv(e) is defined analogously. The vertex PI index defined as the sum of the nu(e) and nv(e). Here the sum is taken over all edges of G. The energy of a graph is defined as the sum of the eigenvalues of adjacency matrix of G and the Laplacian energy of a graph is defined as the sum of the absolute value of difference of laplacian eigenvalues and average degree of G. In theoretical chemistry, the π-electron energy of a conjugated carbon molecule, computed using the Hückel theory, coincides with the energy. Hence results on graph energy assume special significance. The Laplacian matrix of a graph G weighted by the vertex PI weighting is the Laplacian vertex PI matrix and the Laplacian vertex PI eigenvalues of a connected graph G are the eigenvalues of its Laplacian vertex PI matrix. In this study, Laplacian vertex PI energy of a graph is defined of G. We also give some bounds for the Laplacian vertex PI energy of graphs in terms of vertex PI index, the sum of the squares of entries in the Laplacian vertex PI matrix and the absolute value of the determinant of the Laplacian vertex PI matrix. Downloads 154
##### 1999 BIASS in the Estimation of Covariance Matrices and Optimality Criteria

Authors: Juan M. Rodriguez-Diaz

Abstract:

The precision of parameter estimators in the Gaussian linear model is traditionally accounted by the variance-covariance matrix of the asymptotic distribution. However, this measure can underestimate the true variance, specially for small samples. Traditionally, optimal design theory pays attention to this variance through its relationship with the model's information matrix. For this reason it seems convenient, at least in some cases, adapt the optimality criteria in order to get the best designs for the actual variance structure, otherwise the loss in efficiency of the designs obtained with the traditional approach may be very important. Downloads 331
##### 1998 FPGA Implementation of Novel Triangular Systolic Array Based Architecture for Determining the Eigenvalues of Matrix

Abstract:

In this paper, we have presented a novel approach of calculating eigenvalues of any matrix for the first time on Field Programmable Gate Array (FPGA) using Triangular Systolic Arra (TSA) architecture. Conventionally, additional computation unit is required in the architecture which is compliant to the algorithm for determining the eigenvalues and this in return enhances the delay and power consumption. However, recently reported works are only dedicated for symmetric matrices or some specific case of matrix. This works presents an architecture to calculate eigenvalues of any matrix based on QR algorithm which is fully implementable on FPGA. For the implementation of QR algorithm we have used TSA architecture, which is further utilising CORDIC (CO-ordinate Rotation DIgital Computer) algorithm, to calculate various trigonometric and arithmetic functions involved in the procedure. The proposed architecture gives an error in the range of 10−4. Power consumption by the design is 0.598W. It can work at the frequency of 900 MHz. Downloads 327
##### 1997 Normalized Laplacian Eigenvalues of Graphs

Authors: Shaowei Sun

Abstract:

Let G be a graph with vertex set V(G)={v_1,v_2,...,v_n} and edge set E(G). For any vertex v belong to V(G), let d_v denote the degree of v. The normalized Laplacian matrix of the graph G is the matrix where the non-diagonal (i,j)-th entry is -1/(d_id_j) when vertex i is adjacent to vertex j and 0 when they are not adjacent, and the diagonal (i,i)-th entry is the di. In this paper, we discuss some bounds on the largest and the second smallest normalized Laplacian eigenvalue of trees and graphs. As following, we found some new bounds on the second smallest normalized Laplacian eigenvalue of tree T in terms of graph parameters. Moreover, we use Sage to give some conjectures on the second largest and the third smallest normalized eigenvalues of graph. Downloads 246

Authors: Lina Pan

Abstract:

In non-Gaussian clutter of a spherically invariant random vector, in the cases that a certain estimated covariance matrix could become singular, the adaptive target detection of high-range-resolution radar is addressed. Firstly, the restricted maximum likelihood (RML) estimates of unknown covariance matrix and scatterer amplitudes are derived for non-Gaussian clutter. And then the RML estimate of texture is obtained. Finally, a novel detector is devised. It is showed that, without secondary data, the proposed detector outperforms the existing Kelly binary integrator. Downloads 379
##### 1995 An Improved Adaptive Dot-Shape Beamforming Algorithm Research on Frequency Diverse Array

Authors: Yanping Liao, Zenan Wu, Ruigang Zhao

Abstract:

Frequency diverse array (FDA) beamforming is a technology developed in recent years, and its antenna pattern has a unique angle-distance-dependent characteristic. However, the beam is always required to have strong concentration, high resolution and low sidelobe level to form the point-to-point interference in the concentrated set. In order to eliminate the angle-distance coupling of the traditional FDA and to make the beam energy more concentrated, this paper adopts a multi-carrier FDA structure based on proposed power exponential frequency offset to improve the array structure and frequency offset of the traditional FDA. The simulation results show that the beam pattern of the array can form a dot-shape beam with more concentrated energy, and its resolution and sidelobe level performance are improved. However, the covariance matrix of the signal in the traditional adaptive beamforming algorithm is estimated by the finite-time snapshot data. When the number of snapshots is limited, the algorithm has an underestimation problem, which leads to the estimation error of the covariance matrix to cause beam distortion, so that the output pattern cannot form a dot-shape beam. And it also has main lobe deviation and high sidelobe level problems in the case of limited snapshot. Aiming at these problems, an adaptive beamforming technique based on exponential correction for multi-carrier FDA is proposed to improve beamforming robustness. The steps are as follows: first, the beamforming of the multi-carrier FDA is formed under linear constrained minimum variance (LCMV) criteria. Then the eigenvalue decomposition of the covariance matrix is ​​performed to obtain the diagonal matrix composed of the interference subspace, the noise subspace and the corresponding eigenvalues. Finally, the correction index is introduced to exponentially correct the small eigenvalues ​​of the noise subspace, improve the divergence of small eigenvalues ​​in the noise subspace, and improve the performance of beamforming. The theoretical analysis and simulation results show that the proposed algorithm can make the multi-carrier FDA form a dot-shape beam at limited snapshots, reduce the sidelobe level, improve the robustness of beamforming, and have better performance. Downloads 49
##### 1994 A Variant of a Double Structure-Preserving QR Algorithm for Symmetric and Hamiltonian Matrices

Authors: Ahmed Salam, Haithem Benkahla

Abstract:

Recently, an efficient backward-stable algorithm for computing eigenvalues and vectors of a symmetric and Hamiltonian matrix has been proposed. The method preserves the symmetric and Hamiltonian structures of the original matrix, during the whole process. In this paper, we revisit the method. We derive a way for implementing the reduction of the matrix to the appropriate condensed form. Then, we construct a novel version of the implicit QR-algorithm for computing the eigenvalues and vectors. Downloads 305
##### 1993 Using Eigenvalues and Eigenvectors in Population Growth and Stability Obtaining

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The Knowledge of the population growth of a nation is paramount to national planning. The population of a place is studied and a model developed over a period of time, Matrices is used to form model for population growth. The eigenvalue ƛ of the matrix A and its corresponding eigenvector X is such that AX = ƛX is calculated. The stable age distribution of the population is obtained using the eigenvalue and the characteristic polynomial. Hence, estimation could be made using eigenvalues and eigenvectors. Downloads 383
##### 1992 Performance Comparison of Wideband Covariance Matrix Sparse Representation (W-CMSR) with Other Wideband DOA Estimation Methods

Authors: Sandeep Santosh, O. P. Sahu

Abstract:

In this paper, performance comparison of wideband covariance matrix sparse representation (W-CMSR) method with other existing wideband Direction of Arrival (DOA) estimation methods has been made.W-CMSR relies less on a priori information of the incident signal number than the ordinary subspace based methods.Consider the perturbation free covariance matrix of the wideband array output. The diagonal covariance elements are contaminated by unknown noise variance. The covariance matrix of array output is conjugate symmetric i.e its upper right triangular elements can be represented by lower left triangular ones.As the main diagonal elements are contaminated by unknown noise variance,slide over them and align the lower left triangular elements column by column to obtain a measurement vector.Simulation results for W-CMSR are compared with simulation results of other wideband DOA estimation methods like Coherent signal subspace method (CSSM), Capon, l1-SVD, and JLZA-DOA. W-CMSR separate two signals very clearly and CSSM, Capon, L1-SVD and JLZA-DOA fail to separate two signals clearly and an amount of pseudo peaks exist in the spectrum of L1-SVD. Downloads 391
##### 1991 Principle Components Updates via Matrix Perturbations

Abstract:

This paper highlights a new approach to look at online principle components analysis (OPCA). Given a data matrix X R,^m x n we characterise the online updates of its covariance as a matrix perturbation problem. Up to the principle components, it turns out that online updates of the batch PCA can be captured by symmetric matrix perturbation of the batch covariance matrix. We have shown that as n→ n0 >> 1, the batch covariance and its update become almost similar. Finally, utilize our new setup of online updates to find a bound on the angle distance of the principle components of X and its update. Downloads 119
##### 1990 Matrix Valued Difference Equations with Spectral Singularities

Abstract:

In this study, we examine some spectral properties of non-selfadjoint matrix-valued difference equations consisting of a polynomial type Jost solution. The aim of this study is to investigate the eigenvalues and spectral singularities of the difference operator L which is expressed by the above-mentioned difference equation. Firstly, thanks to the representation of polynomial type Jost solution of this equation, we obtain asymptotics and some analytical properties. Then, using the uniqueness theorems of analytic functions, we guarantee that the operator L has a finite number of eigenvalues and spectral singularities. Downloads 367
##### 1989 Random Matrix Theory Analysis of Cross-Correlation in the Nigerian Stock Exchange

Abstract:

In this paper we use Random Matrix Theory to analyze the eigen-structure of the empirical correlations of 82 stocks which are consistently traded in the Nigerian Stock Exchange (NSE) over a 4-year study period 3 August 2009 to 26 August 2013. We apply the Marchenko-Pastur distribution of eigenvalues of a purely random matrix to investigate the presence of investment-pertinent information contained in the empirical correlation matrix of the selected stocks. We use hypothesised standard normal distribution of eigenvector components from RMT to assess deviations of the empirical eigenvectors to this distribution for different eigenvalues. We also use the Inverse Participation Ratio to measure the deviation of eigenvectors of the empirical correlation matrix from RMT results. These preliminary results on the dynamics of asset price correlations in the NSE are important for improving risk-return trade-offs associated with Markowitz’s portfolio optimization in the stock exchange, which is pursued in future work. Downloads 248
##### 1988 Investigation of the Singularity under Non-Proportional Damping in an Analytical Solution of Brake Squeal

Authors: Caiyu Xie, Paul A. Meehan

Abstract:

This research investigates a singularity in a published analytical solution of brake squeal due to mode coupling. Brake squeal is an annoying tonal noise that results from the slowing of a vehicle with friction brakes. Its occurrence in road and rail transportation seems unpredictable, i.e., a ‘squealing’ brake does not squeal during all braking events. The analytical solution predicts squeal accurately, except at a singularity associated with large non-proportional damping and repeated roots. Physically this occurs at the exact transition to stiffness mode coupling, i.e., complex roots (Δ=0). The numerical and analytical results are compared and differences are investigated. The reason for the deviation is identified, and an approximation of the eigenvalues at Δ=0 is made. In particular, the numerical and analytical solutions under non-proportional damping are produced. Modal mass, stiffness and damping matrices based on the analytical eigenvectors are formulated and the values of which are compared with that from numerical results. By taking the limit of the eigenvalue expressions at Δ=0, it is found that the analytical eigenvalues approach infinity. It is shown that singularity occurs because the modal mass matrix goes to zero at Δ=0. Moreover, the approximations of real and imaginary parts of the eigenvalues are plotted against Δμ. It is found that the approximations are closest to the numerical solution when they are the largest or smallest values in the eigenvalues vs. Δμ plot. Since the analytical solution is undefined at Δ=0 when damping is non-proportional, an alternative way to obtain the eigenvalues is investigated. The analytical solution at Δ=0 can be approximated by averaging the eigenvalues that are located on both sides of the critical friction coefficient (the average of eigenvalues at μ𝒸ᵣᵢₜ-Δμ and μ𝒸ᵣᵢₜ+Δμ). In summary, the analytical solution under non-proportional damping is undefined at Δ=0 because of the zero modal mass matrix. Moreover, the eigenvalues at the point where singularity occurs can be approximated by taking the average of the eigenvalues located on both sides of the critical friction coefficient. It is shown that reasonable approximations can be obtained at the critical points in the average eigenvalues vs. Δμ plot. Downloads 4
##### 1987 A Time-Reducible Approach to Compute Determinant |I-X|

Authors: Wang Xingbo

Abstract:

Computation of determinant in the form |I-X| is primary and fundamental because it can help to compute many other determinants. This article puts forward a time-reducible approach to compute determinant |I-X|. The approach is derived from the Newton’s identity and its time complexity is no more than that to compute the eigenvalues of the square matrix X. Mathematical deductions and numerical example are presented in detail for the approach. By comparison with classical approaches the new approach is proved to be superior to the classical ones and it can naturally reduce the computational time with the improvement of efficiency to compute eigenvalues of the square matrix. Downloads 325
##### 1986 Improvement a Lower Bound of Energy for Some Family of Graphs, Related to Determinant of Adjacency Matrix

Abstract:

Let G be a simple graph with the vertex set V (G) and with the adjacency matrix A (G). The energy E (G) of G is defined to be the sum of the absolute values of all eigenvalues of A (G). Also let n and m be number of edges and vertices of the graph respectively. A regular graph is a graph where each vertex has the same number of neighbours. Given a graph G, its line graph L(G) is a graph such that each vertex of L(G) represents an edge of G; and two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G. In this paper we show that for every regular graphs and also for every line graphs such that (G) 3 we have, E(G) 2nm + n 1. Also at the other part of the paper we prove that 2 (G) E(G) for an arbitrary graph G.

Keywords: eigenvalues, energy, line graphs, matching number

##### 1985 A Contribution to the Polynomial Eigen Problem

Authors: Malika Yaici, Kamel Hariche, Tim Clarke

Abstract:

The relationship between eigenstructure (eigenvalues and eigenvectors) and latent structure (latent roots and latent vectors) is established. In control theory eigenstructure is associated with the state space description of a dynamic multi-variable system and a latent structure is associated with its matrix fraction description. Beginning with block controller and block observer state space forms and moving on to any general state space form, we develop the identities that relate eigenvectors and latent vectors in either direction. Numerical examples illustrate this result. A brief discussion of the potential of these identities in linear control system design follows. Additionally, we present a consequent result: a quick and easy method to solve the polynomial eigenvalue problem for regular matrix polynomials. Downloads 361
##### 1984 The Norm, Singular Value and Condition Number Analysis for the Hadamard Matrices

Authors: Emine Tuğba Akyüz

Abstract:

In this study, the analysis of Hadamard matrices, which is a special type of matrix, was made under three headings: norms, singular values, condition number. Six norm types was applied to Hadamard matrices and the relationship between the results and the size of the matrix has been studied. As a result of the investigation when 2-norm was used on the problem Hx =f, the equation ‖x‖_2= ‖f‖_2/√n was shown (H is n-dimensional Hadamard matrix). Related with this, the relationship between the the singular value of H and 2-norm and eigenvalues was shown. Then, the evaluation of condition number for Hx =f was made. Downloads 229
##### 1983 Inverse Scattering for a Second-Order Discrete System via Transmission Eigenvalues

Authors: Abdon Choque-Rivero

Abstract:

The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice when the coefficients are real valued. The inverse problem of recovery of the coefficients from various data sets containing the so-called transmission eigenvalues is analyzed. The Marchenko method is utilized to solve the corresponding inverse problem. Downloads 70
##### 1982 A Combined Error Control with Forward Euler Method for Dynamical Systems

Authors: R. Vigneswaran, S. Thilakanathan

Abstract:

Variable time-stepping algorithms for solving dynamical systems performed poorly for long time computations which pass close to a fixed point. To overcome this difficulty, several authors considered phase space error controls for numerical simulation of dynamical systems. In one generalized phase space error control, a step-size selection scheme was proposed, which allows this error control to be incorporated into the standard adaptive algorithm as an extra constraint at negligible extra computational cost. For this generalized error control, it was already analyzed the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. In this paper, this result was extended to the linear system whose coefficient matrix has complex eigenvalues with negative real parts. Some theoretical results were obtained and numerical experiments were carried out to support the theoretical results. Downloads 247
##### 1981 Survey of Methods for Solutions of Spatial Covariance Structures and Their Limitations

Abstract:

In modelling environment processes, we apply multidisciplinary knowledge to explain, explore and predict the Earth's response to natural human-induced environmental changes. Thus, the analysis of spatial-time ecological and environmental studies, the spatial parameters of interest are always heterogeneous. This often negates the assumption of stationarity. Hence, the dispersion of the transportation of atmospheric pollutants, landscape or topographic effect, weather patterns depends on a good estimate of spatial covariance. The generalized linear mixed model, although linear in the expected value parameters, its likelihood varies nonlinearly as a function of the covariance parameters. As a consequence, computing estimates for a linear mixed model requires the iterative solution of a system of simultaneous nonlinear equations. In other to predict the variables at unsampled locations, we need to know the estimate of the present sampled variables. The geostatistical methods for solving this spatial problem assume covariance stationarity (locally defined covariance) and uniform in space; which is not apparently valid because spatial processes often exhibit nonstationary covariance. Hence, they have globally defined covariance. We shall consider different existing methods of solutions of spatial covariance of a space-time processes at unsampled locations. This stationary covariance changes with locations for multiple time set with some asymptotic properties.

Keywords: parametric, nonstationary, Kernel, Kriging

##### 1980 Closed-Form Sharma-Mittal Entropy Rate for Gaussian Processes

Authors: Septimia Sarbu

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The entropy rate of a stochastic process is a fundamental concept in information theory. It provides a limit to the amount of information that can be transmitted reliably over a communication channel, as stated by Shannon's coding theorems. Recently, researchers have focused on developing new measures of information that generalize Shannon's classical theory. The aim is to design more efficient information encoding and transmission schemes. This paper continues the study of generalized entropy rates, by deriving a closed-form solution to the Sharma-Mittal entropy rate for Gaussian processes. Using the squeeze theorem, we solve the limit in the definition of the entropy rate, for different values of alpha and beta, which are the parameters of the Sharma-Mittal entropy. In the end, we compare it with Shannon and Rényi's entropy rates for Gaussian processes. Downloads 366

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##### 1978 Analysis of Cross-Correlations in Emerging Markets Using Random Matrix Theory

Abstract:

This paper investigates the universal financial dynamics in two dominant stock markets in Sub-Saharan Africa, through an in-depth analysis of the cross-correlation matrix of price returns in Nigerian Stock Market (NSM) and Johannesburg Stock Exchange (JSE), for the period 2009 to 2013. The strength of correlations between stocks is known to be higher in JSE than that of the NSM. Particularly important for modelling Nigerian derivatives in the future, the interactions of other stocks with the oil sector are weak, whereas the banking sector has strong positive interactions with the other sectors in the stock exchange. For the JSE, it is the oil sector and beverages that have greater sectorial correlations, instead of the banks which have the weaker correlation with other sectors in the stock exchange. Downloads 223
##### 1977 Turing Pattern in the Oregonator Revisited

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In this paper, we reconsider the analysis of the Oregonator model. We highlight an error in this analysis which leads to an incorrect depiction of the parameter region in which diffusion driven instability is possible. We believe that the cause of the oversight is the complexity of stability analyses based on eigenvalues and the dependence on parameters of matrix minors appearing in stability calculations. We regenerate the parameter space where Turing patterns can be seen, and we use the common Lyapunov function (CLF) approach, which is numerically reliable, to further confirm the dependence of the results on diffusion coefficients intensities. Downloads 277
##### 1976 The Influence of Covariance Hankel Matrix Dimension on Algorithms for VARMA Models

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Some estimation methods for VARMA models, and Multivariate Time Series Models in general, rely on the use of a Hankel matrix. It is known that if the data sample is populous enough and the dimension of the Hankel matrix is unnecessarily large, this may result in an unnecessary number of computations as well as in numerical problems. In this sense, the aim of this paper is two-fold. First, we provide some theoretical results for these matrices which translate into a lower dimension for the matrices normally used in the algorithms. This contribution thus serves to improve those methods from a numerical and, presumably, statistical point of view. Second, we have chosen an estimation algorithm to illustrate in practice our improvements. The results we obtained in a simulation of VARMA models show that an increase in the size of the Hankel matrix beyond the theoretical bound proposed as valid does not necessarily lead to improved practical results. Therefore, for future research, we propose conducting similar studies using any of the linear system estimation methods that depend on Hankel matrices. Downloads 162
##### 1975 Theoretical Framework for Value Creation in Project Oriented Companies

Authors: Mariusz Hofman

Abstract:

The paper ‘Theoretical framework for value creation in Project-Oriented Companies’ is designed to determine, how organisations create value and whether this allows them to achieve market success. An assumption has been made that there are two routes to achieving this value. The first one is to create intangible assets (i.e. the resources of human, structural and relational capital), while the other one is to create added value (understood as the surplus of revenue over costs). It has also been assumed that the combination of the achieved added value and unique intangible assets translates to the success of a project-oriented company. The purpose of the paper is to present hypothetical and deductive model which describing the modus operandi of such companies and approach to model operationalisation. All the latent variables included in the model are theoretical constructs with observational indicators (measures). The existence of a latent variable (construct) and also submodels will be confirmed based on a covariance matrix which in turn is based on empirical data, being a set of observational indicators (measures). This will be achieved with a confirmatory factor analysis (CFA). Due to this statistical procedure, it will be verified whether the matrix arising from the adopted theoretical model differs statistically from the empirical matrix of covariance arising from the system of equations. The fit of the model with the empirical data will be evaluated using χ2, RMSEA and CFI (Comparative Fit Index). How well the theoretical model fits the empirical data is assessed through a number of indicators. If the theoretical conjectures are confirmed, an interesting development path can be defined for project-oriented companies. This will let such organisations perform efficiently in the face of the growing competition and pressure on innovation. Downloads 179
##### 1974 Conditions on Expressing a Matrix as a Sum of α-Involutions

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Let F be C or R, where C and R are the set of complex numbers and real numbers, respectively, and n be a natural number. An n-by-n matrix A over the field F is called an α-involutory matrix or an α-involution if there exists an α in the field such that the square of the matrix is equal to αI, where I is the n-by-n identity matrix. If α is a complex number or a nonnegative real number, then an n-by-n matrix A over the field F can be written as a sum of n-by-n α-involutory matrices over the field F if and only if the trace of that matrix is an integral multiple of the square root of α. Meanwhile, if α is a negative real number, then a 2n-by-2n matrix A over R can be written as a sum of 2n-by-2n α-involutory matrices over R if and only the trace of the matrix is zero. Some other properties of α-involutory matrices are also determined Downloads 90
##### 1973 Signs-Only Compressed Row Storage Format for Exact Diagonalization Study of Quantum Fermionic Models

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The present paper describes a high-performance parallel realization of an exact diagonalization solver for quantum-electron models in a shared memory computing system. The proposed algorithm contains a storage format for efficient computing eigenvalues and eigenvectors of a quantum electron Hamiltonian matrix. The results of the test calculations carried out for 15 sites Hubbard model demonstrate reduction in the required memory and good multiprocessor scalability, while maintaining performance of the same order as compressed row storage. Downloads 272