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In this article, a continuous-time complex-valued projection neural network (CCPNN) in a matrix state space is first proposed for a general complex-variable basis pursuit problem. The proposed CCPNN is proved to be stable in the sense of Lyapunov and to be globally convergent to the optimal solution under the condition that the sensing matrix is not row full rank. Furthermore, an improved discrete-time complex projection neural network (IDCPNN) is proposed by discretizing the CCPNN model. The proposed IDCPNN consists of a two-step stop strategy to reduce the calculational cost. The proposed IDCPNN is theoretically guaranteed to be global convergent to the optimal solution. Finally, the proposed IDCPNN is applied to the reconstruction of sparse signals based on compressed sensing. Computed results show that the proposed IDCPNN is superior to related complex-valued neural networks and conventional basis pursuit algorithms in terms of solution quality and computation time.

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

This paper addresses the robust stability and stabilization problems for fractional-order systems with polytopic uncertainties. By introducing homogeneous polynomial parameter-dependent matrix form...

Existing Dictionary Learning and Sparse Coding (DLSC) algorithms for Symmetric Positive Definite (SPD) matrices usually adopt Reproducing Kernel Hilbert Space as workspace to perform necessary linear operations. But those methods heavily rely on ideal kernel maps and they are lack of robustness when facing different SPD data, especially in the case of high-sparsity coding. Different from existing methods, we explore a new workspace called Symmetric Matrices Inner Product Space (SMIPS) for modeling robust DLSC of SPD matrices. SMIPS, which can be treated as the minimal linear expansion space of SPD manifold, is a linear space equipped with a pre-defined inner product so it supports linear operations. Modelling DLSC in SMIPS is more intuitive, and SPD data can preserve matrix form in SMIPS. Then, in this paper, a Riemannian Geometry Preserving (RGP) method based on graph-regularized is proposed to include the Riemannian geometric information of original SPD data, so that improves the discriminability of sparse codes, and a sparse coding algorithm is developed to acquire sparse codes of SPD matrices in SMIPS efficiently, which extend the feature-sign search to such matrix space. For dictionary learning, an overall learning strategy utilizing the closed form solution of RGP codes is proposed to speed up the iterative dictionary learning process. Experiments on several computer vision tasks show that our algorithm is more robust even with high-sparsity coding across different datasets and outperforms other comparative algorithms in terms of recognition accuracy and learning efficiency, which also demonstrates the validity of SMIPS as a workspace.

This paper considers the minimax design of two-dimensional (2D) finite impulse response (FIR) half-band filters. First, the design problem is formulated in a matrix form, where the half-band constraints are expressed as a pair of matrix equations. By matrix transformations, the constrained minimax problem is transformed into an unconstrained one. Then, we propose an efficient iterative reweighted least squares (IRLS) algorithm to solve this problem. The weighted least squares (WLS) subproblems arising from the IRLS algorithm are solved using a generalized conjugate gradient (GCG) algorithm. Moreover, the GCG algorithm is guaranteed to converge in a finite number of iterations. In the proposed algorithm, the design coefficients of filters are solved in their matrix form, leading to a great saving in computations and memory space. Design examples and comparisons with existing methods are provided to demonstrate the effectiveness and efficiency of the proposed algorithm.

We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].

A complete mathematical model for measuring the parameters of the scattering matrix for a measurement object [Sx] in the form of a four-port network is developed, in which the mathematical model of the eight-port error is described by 16 parameters of the scattering matrix [E]. In addition, in comparison with the 12-parameter model of the eight-port error network, four parameters are included, which allow taking into account leaks, parasitic transmissions of microwave modules under study (microwave microassemblies). Due to the use of matrix analysis methods, the equations in matrix form are obtained that connect the matrices of the measurement object [Sи] and the actual values of the matrix parameters [Sx], with the aim of enabling the solution of these equations instead of the scattering matrix [E] to use a transmition matrix [T] in the form of cellular matrices [Taa], [Tab], [Tba], [Tbb].

We explore a case example of hybridity between a large worker-owned cooperative and a union through three lenses: organizational forms, multiple institutional logics, and organizational identity. We delineate three types of organizational hybridity: (1) stretching an existing organizational form; (2) creating a new organizational form; and (3) and retaining multiple discrete organizational forms in a common venture. The cooperative–union hybrid shares members from the two contributing organizations, and so can be classified as a matrix sub-form of multi-organizational hybridity. This study describes how the coop-union hybrid manages the multiple logics and identities retained from both contributing organizations. It considers the hazards of combining these logics and identities, and offers some suggestions on how to avoid potential difficulties. Finally, given the complexity and inefficiencies of the matrix form, we explore whether matrix hybridity is a transitional or permanent form in this particular instance of a cooperative–union venture.

The objective in this research is to apply the recurrence neural network for car manufacturing with economy sentiment. The process applies two parallel mathematical methods from matrix approach and graph representation. First the randomly changes in production lines are studied regardless of economy sentiment. Secondly the change of production lines is considered in the case of changing economy sentiment. Thirdly the combination of the two processes in matrix form and graph form is constructed. It is seen that the graph, nodes and recursive process approach will lead to the recurrent neural network as timely sequence process. The development of the matrix conversion to graph representation is done and explained in figures in detail.

espanolEn este articulo, se muestra una aplicacion y avance al metodo que permite encontrar de manera simple los coeficientes constantes que se requieren en la solucion de una ecuacion diferencial con la estructura de Cauchy-Euler, presentada y dada a conocer en [1]. Aqui veremos la forma de construccion de los polinomios caracteristicos, utilizando el metodo mencionado anteriormente como base; con una serie de ecuaciones, matrices y metodos novedosos para resolver este tipo de EDO y, sobre todo, muy practicas para la solucion de la ecuacion de orden superior. La idea es presentar una estructura o metodologia en forma matricial que permita resolver de manera practica la ecuacion diferencial de orden superior con la estructura de Cauchy-Euler. EnglishIn this article application and advance is given to the method that allows to find in a simple way the constant coefficients that are required in the solution of a differential equation with the structure of Cauchy-Euler, presented and made known in [1]. Here we will see the construction form of the characteristic polynomials, using the aforementioned method as a basis; with a series of equations, matrices and novel methods to solve this type of ODEs and, above all, very practical for the equation of a much higher order. The idea is to present a structure or methodology in matrix form that allows to solve in a practical way differential equation of higher order with Cauchy-Euler structure.

Equations of the theory of steady-state complex reactions are considered in matrix form. A set of stage stationarity equations is given, and an algorithm is described for deriving the canonic set of stationarity equations with appropriate corrections for the existence of fast stages in a mechanism. A formula for calculating the number of key compounds is presented. The applicability of the Gibbs rule to estimating the number of independent compounds in a complex reaction is analyzed. Some matrix equations relating the rates of dependent and key substances are derived. They are used as a basis to determine the general diffusion stoichiometry relationships between temperature, the concentrations of dependent reaction participants, and the concentrations of key reaction participants in a catalyst grain. An algorithm is described for calculating heat and mass transfer in a catalyst grain with respect to arbitrary complex heterogeneous catalytic reactions.

Fast design of two-dimensional FIR filters in the least $${l}_{p}$$ -norm sense is investigated in this brief. The design problem is first formulated in a matrix form and then solved by a matrix-based iterative reweighted least squares algorithm. The proposed algorithm includes two loops: one for updating the weighting function and the other for solving the weighted least squares (WLS) subproblems. These WLS subproblems are solved using an efficient matrix-based WLS algorithm, which is an iterative procedure with its initial iterative matrix being the solution matrix in the last iteration, resulting in a considerable CPU-time saving. Through analysis, the new algorithm is shown to have a lower complexity than existing methods. Three design examples are provided to illustrate the high computational efficiency and design precision of the proposed algorithm.

Purpose The purpose of this paper is to obtain an iterative algorithm to find the solution of the coupled Sylvester-like matrix equations. Design/methodology/approach In this work, the matrix form of the conjugate direction (CD) algorithm to find the solution X of the coupled Sylvester-like matrix equations: {A1XB1+M1f1(X)N1=F1,A2XB2+M2f2(X)N2=F2,with fi(X) = X, fi(X) = X¯, fi(X) = XT and fi(X) = XH for i = 1; 2 has been established. Findings It is proven that the algorithm converges to the solution within a finite number of iterations in the absence of round-off errors. Finally, four numerical examples were used to test the proficiency and convergence of the established algorithm. Originality/value The numerical examples have led the author to believe that the generalized CD (GCD) algorithm is efficient and it converges more rapidly in comparison with the CGNR and CGNE algorithms.

In this paper, we have described the optical model of light field in vision correcting display providing clear image to farsighted vision at a close viewing distance. Relationship between display and retina images was expressed to the matrix form based on the optical model of light transportation. We reconstructed retina image by optimizing the display image for the given matrix relationship. By analyzing the image quality of corrected image according to an optical gap between pin-hole and pixel arrays, we obtained the proper optical gap maximizing the correction of farsighted vision in our design condition.

We have developed a method for prediction of antimicrobial resistance by representing the antibiogram and associated metadata in the matrix form, determining parameter of matrix factorization, i.e., the number of the latent factors, carrying out matrix factorization of the initial association matrix and construction of the new association matrix containing predictions of the associations between multiple samples antibiogram and associated metadata.

We study a robust emergency system in a transport network where it is necessary to solve a problem of finding p service centers for randomly emergency demands in nodes. The objective of our problem is to minimize the upper bound of partial object functions for given number of detrimental scenarios. Transport network is modelled in a matrix form i.e elements of the matrix are response times from a potential service center to a location of an user. Such matrix is given for every detrimental scenario. At last every location is given a number of shared users. We present modified micro-genetic algorithm on the instance of transport network of Slovak Republic. We verify it in selected regions of network via comparing it to known exact solutions. We also seek an approximate solution for a total network.

In this paper, a new filter synthesis technology with frequency-dependent coupling (FDC) has been presented by analytical synthesis method instead of exploiting optimization method. Firstly, according to the assigned frequency-independent coupling filter topology, the initial frequency-independent coupling matrix can be synthesized to realize the desired indicators. Then, the expected FDC filter topology should be chosen and the corresponding FDC matrix form can be determined. Finally, and importantly, the obtained FDC matrix for FDC filter topology can be suitably transformed by means of scaling and rotations of coupling and capacitance matrices for realizing the frequency-independent coupling matrix that the form should be consistent with the initial one. To maintain the identical indicators, the frequency-independent coupling matrix which is transformed from the FDC matrix should be equal to the initial synthesized frequency-independent coupling matrix. Hence, the elements in coupling and capacitance matrices can be solved and FDC matrix can be eventually obtained. Above all, the FDC filter synthesis approach has been illustrated with a fifth-order filter example, and the good agreements between the different filter responses have validated the validation of the proposed synthesis method.

This chapter presents a deadlock-avoidance matrix-based supervisory controller for discrete event systems. The formulation of this matrix-based controller makes it direct to write it down from standard manufacturing tools such as the bill of materials or the assembly tree. It is shown that the discrete event controller's matrix form equations plus the Petri Net marking transition equation together provide a complete dynamical description of discrete event systems.

This chapter explains that straightforward cardinal elements doublets and multiplets can be obtained by multiplying the appropriate transfer metrics. The coefficients of the aberration polynomials of the multiplet can be expressed in terms of those of the separate lenses. However, it requires introducing column vectors in an arbitrary pair of conjugate planes in the object plane, with a similar expression for u m , the corresponding vector in the image plane, magnification M.

We show that the r-by-r minors of a d-by-n matrix of variables form a tropical basis of the ideal they generate if and only if one of the following conditions holds: (i) r ⩽ 3 , (ii) r = min { d , n } , (iii) r = 4 and min { d , n } ⩽ 6 . This answers a question raised by M. Chan, A. Jensen, and E. Rubei.

We present an analytical method for the derivation of the return ratio in linear circuits with multiple dependent sources. We show how to find the matrix return ratio from the circuit, by using a superposition of test and input sources, and how to obtain the closed-loop behavior using a linear fractional transformation. The return ratios and closed-loop transfer function are presented in matrix form. The suggested procedure does not require one to solve the circuit; instead, the components of the relevant matrices are computed directly. Deriving the return ratio in matrix form allows us to apply standard tools to study the closed-loop behavior of the circuit and the influence of the components on the latter.

An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and ( R , S )-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.

In everyday life, especially in the electrical circuit, there are many usage of matrix. One use of a matrix is found in the system of linear equations. In the field of electrical circuits there are also problems involving systems of linear equations in matrix form. To solve the system of linear equations in matrix form, in addition to using elementary row operations, also used matlab.

This paper presents the graphics processing unit (GPU) accelerated fundamental alternating-direction-implicit flnite-difierence time-domain (FADI-FDTD) with complex frequency shifted convolu- tional perfectly matched layer (CFS-CPML). The compact matrix form of the conventional ADI-FDTD method with CFS-CPML is formu- lated into FADI-FDTD with its right-hand-sides free of matrix oper- ators, resulting in simpler and more concise update equations. Using Compute Unifled Device Architecture (CUDA), the FADI-FDTD with CFS-CPML is further incorporated into the GPU to exploit data par- allelism. Numerical results show that a much higher e-ciency gain of up to 15 times can be achieved.

This article introduced the research of writing based on the robot personification in 6B matrix form. The 6B matrix form is a new description method of Chinese character. In 6B matrix form, the stroking order (vector) when writing a Chinese character is included in a characteristic 6B matrix, which can be used to realize the personification writing through actuating software.

In this paper, global asymptotic stability is discussed for neural networks with time-varying delay. Several new criteria in matrix inequality form are given to ascertain the uniqueness and global asymptotic stability of equilibrium point for neural networks with time-varying delay based on Lyapunov method and Linear Matrix Inequality (LMI) technique. The proposed LMI approach has the advantage of considering the difference of neuronal excitatory and inhibitory efforts, which is also computationally efficient as it can be solved numerically using recently developed interior-point algorithm. In addition, the proposed results generalize and improve previous works. The obtained criteria also combine two existing conditions into one generalized condition in matrix form. An illustrative example is also given to demonstrate the effectiveness of the proposed results.

In computer-aided design it is both convenient and practical to use the matrix form in representing parametric curves and surfaces. One of the reasons is that the matrix notation allows an easy conversion between different shape representations and provides a convenient implementation in either hardware or software with available matrix facilities. In this work we propose a very simple and efficient procedure for computing in a recursive way the conversion matrices which provide the direct transformation between uniform B-spline and Bézier representations of arbitrary degrees.

This paper presents a linear algebraic procedure in obtaining (6−n) reciprocal screws from an n-screw system of a robotic manipulator. The procedure starts by formulating the reciprocal relationship between a screw system and its reciprocal screw system in matrix form, and augmenting the screw matrix which is assembled with n screws in the screw system. Vectors of cofactors of the augmented row are then produced and the partitioning of the screw matrix is implemented to generate each of the (6−n) reciprocal screws. The theory developed in this paper provides a novel and simple procedure to obtain reciprocal screws from any screw system. The paper illustrates applications to cases with different screw systems. © 2003 Wiley Periodicals, Inc.

This paper describes work undertaken to define the photonics up-skilling capability of higher education (HE) and further education (FE) institutions in Wales. The expertise was compiled in matrix form and included specification of the Training,Research,Equipment and Expertise (TREE) of the relevant institutions. The information contained in the CAMPUS TREEs is designed for use by industry and commerce.

We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners., Comment: 26 pages, 3 figures

An algorithm based on latent variables is introduced for three-way/second-order calibration. Unfolding the matrix calibration data leads to the possibility of applying classical partial least-squares, a popular regression technique in the framework of first-order calibration. The achievement of the second-order advantage is left to a postcalibration procedure called residual bilinearization (RBL), which processes the test samples in the original matrix form, efficiently separating the contribution of the potential interferents from those of the calibrated analytes. The resulting unfolded partial least-squares/RBL algorithm shows a great flexibility, and is able to cope with some data sets deviating from trilinearity.

This paper describes a compact matrix formulation for the steady-state analysis of structural-acoustic systems. A new approach to the problem is adopted that uses impedance and mobility methods commonly found in the analysis of purely structural or purely acoustic systems. The advantages of the approach are that an investigation into the coupling between the structural and acoustic systems is made easier, and it facilitates improved physical insight into the behaviour of structural-acoustic systems. In addition, because the equations describing the complete system are in matrix form, they can be solved easily using a computer. Due to the mismatch of dimensions between structural mobility and acoustic impedance, new terms are introduced for the coupled system analysis; the coupled acoustic impedance and the coupled structural mobility.F-u(force-velocity) andp-Q(pressure-source strength) diagrams are also introduced for impedance and mobility representations of a complete coupled system. Experimental work is presented, in which a simple rectangular acoustic enclosure with five rigid and one flexible side was used, to validate the analytical model and to investigate structural-acoustic coupling.

A formula in matrix form for restoration is given and the matrix with the sampling rate near the Nyquist rate is investigated. Elements of this matrix can be expanded in Laurent series of the sampling rate parameter, which is defined by the quotient of the Nyquist rate and the sampling rate. The Nyquist rate corresponds to a pole. First terms of these Laurent series near the Nyquist rate are given.

In this paper, a matrix form analoguous to the companion matrix which is often encountered in the theory of one dimensional (1-D) linear control systems is suggested for a class of polynomials in two indeterminates and real coefficients, here referred to as two dimensional (2-D) polynomials. These polynomials arise in the context of the theory of 2-D linear discrete control systems. Necessary and sufficient conditions are also presented under which a matrix is equivalent to this companion form. Examples are used to illustrate the ideas developed in this paper.

A computationally efficient iterative algorithm is developed in this paper for the weighted least-squares (WLS) design of two-dimensional (2-D) FIR filters. The new algorithm is based on the optimality condition and uses the matrix iterative technique, which retains the coefficients of 2-D filters in their natural matrix form, resulting in a considerable saving in the amount of computation and memory space. Moreover, two vectors are introduced into the iterative equation of the algorithm, aiming to reduce the number of iterations required for convergence. Some design examples are presented to illustrate the good performance of the proposed algorithm.

Two new approaches for writing kinetic equations in the matrix form or directly in the integrated form are presented here. While the first method allows to derive the kinetic rate matrix of kinetic systems of any kind in a direct and straightforward way, the second approach applies to species that are consumed solely through first order steps, regardless of the complexity of their formation pathways.

Singular value decomposition (SVD) is a powerful linear algebra tool. It is widely used in many different signal processing methods, such principal component analysis (PCA), singular spectrum analysis (SSA), frequency domain decomposition (FDD), subspace identification and stochastic subspace identification method ( SI and SSI ). In each case, the data is arranged appropriately in matrix form and SVD is used to extract the feature of the data set. In this study three different algorithms on signal processing and system identification are proposed: SSA, SSI-COV and SSI-DATA. Based on the extracted subspace and null-space from SVD of data matrix, damage detection algorithms can be developed. The proposed algorithm is used to process the shaking table test data of the 6-story steel frame. Features contained in the vibration data are extracted by the proposed method. Damage detection can then be investigated from the test data of the frame structure through subspace-based and nullspace-based damage indices.

Business intelligence focuses on the discovery of useful retail patterns by combining both historical and prognostic data. Ultimate goal is the orchestration of more targeted sales and marketing efforts. A frequent analytic task includes the discovery of associations between customers and products. Matrix co-clustering techniques represent a common abstraction for solving this problem. We identify shortcomings of previous approaches, such as the explicit input for the number of co-clusters and the common assumption for existence of a block-diagonal matrix form. We address both of these issues and present techniques for automated matrix co-clustering. We formulate the problem as a recursive bisection on Fiedler vectors in conjunction with an eigengap-driven termination criterion. Our technique does not assume perfect block-diagonal matrix structure after reordering. We explore and identify off-diagonal cluster structures by devising a Gaussian-based density estimator. Finally, we show how to explicitly couple co-clustering with product recommendations, using real-world business intelligence data. The final outcome is a robust co-clustering algorithm that can discover in an automatic manner both disjoint and overlapping cluster structures, even in the preserve of noisy observations.

In this paper, the Crank-Nicholson semi-implicit difference scheme is applied to discrete the Rudin-Osher-Fatemi model which was proposed by Rudin et al. in 1992. The semi-implicit discrete scheme avoids the shortcomings of instability and many iterative numbers that the explicit discrete scheme has. We propose an image denoising algorithm in the matrix form. The experimental results show that the new matrix semi-explicit scheme is efficient.

Cultural relics are often damaged and incomplete due to various reasons. For the purpose of helping archaeological studies, we present a novel method for simultaneously restoring the original shapes of a group of similar objects. Based on the assumption that similar shapes are approximately linearly correlated, we use a matrix recovery technique to achieve the restoration. In order to represent input shapes in a matrix form, vectorization of each aligned sample is carried out by stacking coordinates of dense corresponding points that are generated by a surface matching scheme using non-rigid deformation. An experiment using 3D scans of facial sculptures from Bayon is conducted, and the result verifies the feasibility and effectiveness of our method.

IML stands for Interactive Matrix Language and is a matrix language written as a component of the SAS System. IML allows the user to enter numbers and other characters in matrix form and perform data manipulation, functions and algorithms commonly associated with matrices.

After identifying the places in which the palletization of auto batteries from S.C. ROMBAT S.A. Bistrita can be done, the structural kinematic scheme of the articulated robot IGM RT 330.1/370.1 (which can be found in S.C. ROMBAT S.A.) was developed. Forwards, by using the finite displacements theory of a rigid solid, the paper presents an algorithm of calculation in order to establish the relations of dependency between the initial and the final positions of the batteries and the constructive parameters of the robot. The algorithm of calculation has a matrix form that is suitable for computer implementation.

In this work, we study Joachimsthal’s theorems in semi-Euclidean spaces E ν n + 1 . We obtain a relation between the curvatures of the strips in semi-Euclidean spaces E ν n + 1 in matrix form depending on a semi-orthogonal matrix. This matrix equation gives us a generalization of the original Joachimsthal theorem in semi-Euclidean space E 1 3 and the well-known Joachimsthal theorem of the surface theory. In addition, we reobtain the well-known Joachimsthal theorem from our matrix equation in the case n = 2 , ν = 0 and the original Joachimsthal theorem from our matrix equation in the case n = 2 , ν = 1 .

A neutrino mass matrix model M_\nu with M_\nu^T =M_\nu and a model with its inverse matrix form \widetilde{M}_\nu = m_0^2 (M_\nu^*)^{-1} can be diagonalized by the same mixing matrix U_\nu. It is investigated whether a scenario which provides a matrix model M_\nu with normal mass hierarchy can also give a model with an inverted mass hierarchy by considering a model with an inverse form of M_\nu., Comment: 6 pages, 4 figures

This paper is the second of two in which a methodology is presented that provides a structured way of solving Dimensioning and Tolerancing (D&T) problems on new product design. In the previous paper, a methodology was developed for representing the D&T problems of a product in a matrix form known as a Dimensional Requirements/Dimensions (DR/D) matrix. In this paper a comprehensive strategy is presented for determining the values of dimensions and tolerances by satisfying all the relationships represented in DR/D matrix. The comprehensive strategy presented here includes: strategy for separating D&T problems into groups, for determining an optimum solution order for coupled functional equations and tolerance allocation strategies for solving different types of D&T problems. A number of commonly used cost minimization strategies such as the use of standard parts, preferred sizes, preferred fits, and preferred tolerances have also been incorporated into the proposed methodology. The methodology is interactive and suitable for use in a Concurrent Engineering environment.

The authors consider the problem of developing VLSI signal processors for computing convolutions. Convolutions can be efficiently computed by VLSI processors that consist of arrays of adders when they are stated in terms of matrices with elements consisting of only 1, 0, or -1. Unfortunately, when stated in matrix form the published algorithms have matrices with elements other than 1, 0, or -1. The authors explore why this occurs and show how it can be prevented when an algorithm is developed. If this fails, they propose a technique for addressing this problem that consists of replacing each such matrix by the product of two or more matrices whose elements are 1, 0, or -1. >

Expanded from the thermal characteristics of single chip packages, a matrix form thermal characteristic for MCM and MCPs is introduced in this paper. The validity of this characteristic is investigated through a few most commonly used MCM, MCP assembly configurations and mounting boards. A detail discussion is made on the relationship between the thermal characteristic, /spl theta/ matrix and its thermal circuit representation. In addition, two techniques are introduced to extract the equivalent thermal network from a given /spl theta/ matrix or directly from numerical modeling. The extracted thermal network is also tested by numerical examples and has proved to be accurate enough for higher system level applications.

In this paper we use the tensor product notation as the framework of a programming methodology for designing various parallel prefix algorithms. In this methodology, we first express a computational problem in its matrix form. Next, we formulate a matrix equation for the matrix of the computational problem. Then, solve the matrix equation to obtain some simple matrices. Finally, we recursively factorize the subproblem to obtain a tensor product formula representing an algorithm for this problem. We will use the parallel prefix computation problem to illustrate our methodology and derive various parallel prefix algorithms including divide-and-conquer and recursive doubling algorithms.

A class of dynamic systems is described by linear differential equations: $$ \mathop {{x_i}}\limits^.= \sum\limits_{l = 1}^n {{A_{il}}{x_l} + \sum\limits_{\theta= 1}^m {{B_{i\theta }}{u_\theta }} ,\quad i = 1,,n} $$ (7.1) where x i (t) are the state variables and u s (t) are the control inputs of the system. A il and B is are coefficients which are either constant or are functions of time. Often, the nonlinear dynamics of Eq. (4.1) is linearized to Eq. (7.1) using Taylor’ s expansion valid in a small region of the state space. In a matrix form, the state Eqs. (7.1) can be rewritten as $$ \mathop {\overline x }\limits^.= A\overline x+ B\overline u $$ (7.2) where A is a (n × n) matrix with entries A il , Bisa (n × m) matrix with elements B is ,is a (n × 1) vector with ith element x i , and ū is a (m × 1) vector with ith element u i .

Gene action and interaction have been defined in the literature by the use of a variety of notations (Mather 1949; Hayman 1954, 1955, 1957; Jinks 1954; Kempthorne 1954, 1955). This leads to unnecessary complications in understanding the subject. This paper provides a simple convenient way of translating one parameterization into another and illustrates the simple linear relationship between them. The various notations are written in matrix form by the use of a specification matrix. This provides a convenient compact method of presentation of the relevant Equations. The linear relationship between the genetic parameters enables these to be estimated in the most convenient way and then converted to other parameters for the purposes of comparison. The generation means Equations of Hayman (1958) are derived using the matrix formulation as an illustration of the use of the specification matrix.