Matrix Boundary Value Problem on Hyperbola

We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only one. Secondly, a special boundary value problem of lower triangular matrix is presented and transformed into four related boundary value problems. Finally, Liouville theorem and Painlevé theorem and pseudo-orthogonal polynomials are used to give solutions.

Werner-Type Matrix Valued Rational Interpolation and Its Recurrence Algorithms

In this paper, a practical Werner-type continued fraction method for solving matrix valued rational interpolation problem is provided by using a generalized inverse of matrices. In order to reduce the continued fraction form to rational function form of the interpolants, an efficient forward recurrence algorithm is obtained.

A SEMI-CONJUGATE MATRIX BOUNDARY VALUE PROBLEM FOR GENERAL ORTHOGONAL POLYNOMIALS ON AN ARBITRARY SMOOTH JORDAN CURVE

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.

PROBLEMS AND METHODS IN MATRIX VALUED RATIONAL INTERPOLATION

A variety of matrix rational interpolation problems include the partial realizationproblem for matrix power series and the minimal rational interpolation problem for generalmatrix functions.Several problems in circuit theory and digital filter design can also be re-duced to the solution of matrix rational interpolation problems[1—4].By means of thereachability and the observability indices of defined pairs of matrices,Antoulas,Ball,Kang and Willems solved the minimal matrix rational interpolation problem in[1].On

Two Dual Expansions for Generalized Bivariate Thiele-Type Matrix Valued Interpolating Continued Fractions

A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.

REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS

In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.

The εAlgorithm and ηAlgorithm for Generalized InverseRectangular Matrix Pade Approximants

An axiomatic definition for the generalized inverse matrix Pade approximation (GMPA) is introduced. The matrix rational approximants are of the form of the matrix valued numerator and the scalar denominator. By means of generalized inverse for matrices, the ε algorithm for the computation of GMPA is established. The well known Wynn identity for GMPA is proved on the basis of ε algorithm. The η algorithm is defined in a similar way. The equivalence relation between ε algorithm and η algorithm is proposed. Some common examples and a numerical example are given to illustrate the methods in this paper.

Three-Dimensional Generalized Inverse Matrix Rational Interpolation

In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fraction form, with matrix numerator and scalar denominator. Some properties of TGMRI are given. An efficient recursive algorithm is proposed. The results in the paper can be extend to n variable.

Mobility and equilibrium stability analysis of pin-jointed mechanisms with equilibrium matrix SVD

Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equilibrium stability of mechanisms according to the effects of the external loads. The first and second variations of the potential energy function of mechanisms under conservative force field are analyzed. Based on the singular value decomposition (SVD) method, a new crite- rion for the mobility and equilibrium stability of mechanisms can be concluded by analyzing the equilibrium matrix. The mobility and stability of mechanisms can be classified by unified matrix formulae. A number of examples are given to demonstrate the proposed criterion. In the end, criteria are summarized in a table.

Automatic Calculation of Dimension Chains in AutoCAD

In the course of mechanical part designing, process p lanning and assembling designing, we often have to calculate and analyse a dimen sion chain. Traditionally, a dimension chain is established and calculated m anually. With wide computer application in the field of mechanical design and ma nufacture, people began to use a computer to acquire and calculate a dimension c hain automatically. In reported work, a dimension chain can be established and c alculated automatically. However, dimension text values of dimensions composing a dimension chain and these dimensions’ tolerance’s upper values and lower valu es are put into a computer manually, which is inefficient and easy to make mis takes. In order to overcome above difficulties. it is very important to acquir e noted dimensions automatically, furthermore analyse and calculate a dimens ion chain, then show results. At present AutoCAD softwares of Autodesk company h ave been used popularly in mechanical designing. For automatically acquiring noted dimensions, analyzing and calculating a dimension chain in a design draw in AutoCAD, this paper introduces the solvable scheme of automatic dimension acq uisition and dimension chain calculation in AutoCAD by ObjectARX. ObjectARX is a developing tool for AutoCAD. In this paper a dimension chain is expressed b y three matrixes, which respectively stand for dimension text value matrix, tole rance’s upper value matrix and tolerance’s lower value matrix. The developed p rogram can be used to both calculate a assembling dimension chain, and a process dimension chain. When the program running in AutoCAD, noted dimensions comp osing a dimension chain in AutoCAD are selected in turn with a mouse, then the c omputer begin to calculate the dimension chain and results are shown in a dialog box. A running example is given in this paper.

A local f-x Cadzow method for noise reduction of seismic data obtained in complex formations

A noise-reduction method with sliding called the local f-x Cadzow noise-reduction method, windows in the frequency-space （f-x） domain, is presented in this paper. This method is based on the assumption that the signal in each window is linearly predictable in the spatial direction while the random noise is not. For each Toeplitz matrix constructed by constant frequency slice, a singular value decomposition （SVD） is applied to separate signal from noise. To avoid edge artifacts caused by zero percent overlap between windows and to remove more noise, an appropriate overlap is adopted. Besides flat and dipping events, this method can enhance curved and conflicting events. However, it is not suitable for seismic data that contains big spikes or null traces. It is also compared with the SVD, f-x deconvolution, and Cadzow method without windows. The comparison results show that the local Cadzow method performs well in removing random noise and preserving signal. In addition, a real data example proves that it is a potential noise-reduction technique for seismic data obtained in areas of complex formations.

A Geometric View on Inner Transformation between the Variables of a Linear Regression Model

In the teaching and researching of linear regression analysis, it is interesting and enlightening to explore how the dependent variable vector can be inner-transformed into regression coefficient estimator vector from a visible geometrical view. As an example, the roadmap of such inner transformation is presented based on a simple multiple linear regression model in this work. By applying the matrix algorithms like singular value decomposition (SVD) and Moore-Penrose generalized matrix inverse, the dependent variable vector lands into the right space of the independent variable matrix and is metamorphosed into regression coefficient estimator vector through the three-step of inner transformation. This work explores the geometrical relationship between the dependent variable vector and regression coefficient estimator vector as well as presents a new approach for vector rotating.

Research and Design of Coordinated Control Strategy for Smart Electromechanical Actuator System

In order to improve the frequency response and anti-interference characteristics of the smart electromechanical actuator(EMA)system,and aiming at the force fighting problem when multiple actuators work synchronously,a multi input multi output(MIMO)position difference cross coupling control coordinated strategy based on double‑closed-loop load feedforward control is proposed and designed.In this strategy,the singular value method of return difference matrix is used to design the parameter range that meets the requirements of system stability margin,and the sensitivity function and the H_(∞)norm theory are used to design and determine the optimal solution in the obtained parameter stability region,so that the multi actuator system has excellent synchronization,stability and anti-interference.At the same time,the mathematical model of the integrated smart EMA system is established.According to the requirements of point-to-point control,the controller of double-loop control and load feedforward compensation is determined and designed to improve the frequency response and anti-interference ability of single actuator.Finally,the 270 V high-voltage smart EMA system experimental platform is built,and the frequency response,load feedforward compensation and coordinated control experiments are carried out to verify the correctness of the position difference cross coupling control strategy and the rationality of the parameter design,so that the system can reach the servo control indexes of bandwidth 6 Hz,the maximum output force 20000 N and the synchronization error≤0.1 mm,which effectively solves the problem of force fighting.

Function synthesis algorithm based on RTD-based three-variable universal logic gates

Compared with complementary metal–oxide semiconductor(CMOS), the resonant tunneling device(RTD) has better performances; it is the most promising candidate for next-generation integrated circuit devices. The universal logic gate is an important unit circuit because of its powerful logic function, but there are few function synthesis algorithms that can implement an n-variable logical function by RTD-based universal logic gates. In this paper, we propose a new concept, i.e., the truth value matrix. With it a novel disjunctive decomposition algorithm can be used to decompose an arbitrary n-variable logical function into three-variable subset functions. On this basis, a novel function synthesis algorithm is proposed, which can implement arbitrary n-variable logical functions by RTD-based universal threshold logic gates(UTLGs), RTD-based three-variable XOR gates(XOR3s), and RTD-based three-variable universal logic gate(ULG3s). When this proposed function synthesis algorithm is used to implement an n-variable logical function, if the function is a directly disjunctive decomposition one, the circuit structure will be very simple, and if the function is a non-directly disjunctive decomposition one, the circuit structure will be simpler than when using only UTLGs or ULG3s. The proposed function synthesis algorithm is straightforward to program, and with this algorithm it is convenient to implement an arbitrary n-variable logical function by RTD-based universal logic gates.