A definition of column reduced proper rational matrices

Relying on the theory behind the Smith-MacMillan form of a rational matrix at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s = \infty</tex> , a necessary and sufficient condition is derived from the solvability of matrix Diophantine equations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s)X(s)+B(s)Y(s) = M(s)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s), B(s)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M(s)</tex> are given proper rational matrices and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y(s)</tex> are unknown proper rational matrices. It is shown that the above result can be used in order to resolve in a new illuminating way the exact model matching problem (EMMP). If a solution to EMMP exists, then the family of all solutions is parametrized.

: This paper is concerned with the development of a complete set of invariants and canonical forms under feedback for linear systems characterized by proper rational transfer matrices. The invariants are determined in the frequency domain and consist of the Kronecker set of controllability indices together with a canonical form for the numerator of the transfer matrix under the action of a stabilizer subgroup of the unidoular group of polynomial matrices. Techniques used are algebro-geometric in nature. (Author)

Nonregular Static State Feedback for Linear Bond Graph Models

Investigation of finite/infinite unidirectional elastic phononic plates by BEM

By considering a convolutional code as the range space of some linear map over a rational field, we define minimal polynomial bases for a rational vector space as a natural extension of the concept of minimal convolutional encoders. Furthermore, by using the notion of dual spaces, it is shown how relatively straightforward it is to construct a minimal polynomial basis for the direct space. An application of this concept to multivariable linear system theory is also noted by constructing left and right standard matrix factorizations of proper rational matrices which generalize to the multivariable case the classical representation of a proper rational function as a ratio of two relatively prime polynomials with a denominator of degree larger or equal to the one of the numerator.

Qualitative Problems in Matrix Theory

In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix A of size N × M with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix A there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: (1) the matrix A contains at least three different columns; (2) if one places the columns of the matrix A in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has at least one unity to the left of the line and at least one zero to the right of the line. It is established that the response matrix A obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above-mentioned conditions of primitiveness of the square matrix B of order M with the entries \({b_{ij}} = \sum\limits_{\ell = 1}^N {(1 - {a_{\ell i}}){a_{\ell i}}} \) has been proved. Using the matrix analysis, we have proved that the primitiveness of the matrix B ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits.

There are various matrices that are naturally associated with a graph, such as the adjacency matrix, the incidence matrix, and the Laplacian. One of the main problems of algebraic graph theory is to determine precisely how, or whether, properties of graphs are reflected in the algebraic properties of such matrices.

The construction of filters arising from linear neural networks with feed-backward excitatory-inhibitory connections is presented. Spatially invariant coupling between neurons and the distribution of neuron-receptor units in the form of a uniform square grid yield the TBT (Toeplitz-Block-Toeplitz) connection matrix. Utilizing the relationship between spectral properties of such matrices and their generating functions, the method for construction of recurrent linear networks is addressed. By appropriately bounding the generating function, the connection matrix eigenvalues are kept in the desired range allowing for large matrix inverse to be approximated by a convergent power series. Instead of matrix inversion, the single pass convolution with the filter obtained from the network connection weights is applied when solving the network. For the case of inter-neuron coupling in the form of a function that is expandable in a Fourier series in polar angle, the network response filter is shown to be steerable.

We study deterministic one-way communication complexity of functions with Hankel communication matrices. In this paper some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.

Big Ramsey degrees in ultraproducts of finite structures

An algorithm for inverting rational matrices

A special coordinate basis for multivariable linear systems is introduced here. Several fundamental properties of linear systems regarding controllability (stabiliza-bility), observability (detectability), invariant zeros, decoupling zeros, infinite zero structure, effect of feedback on zero structure, squaring down, diagonal and triangular decoupling etc. can be directly displayed in terms of the special coordinate basis. Moreover, connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established.

A special coordinate basis of multivariable linear systems is introduced here. Several fundamental properties of linear systems regarding controllability (stabilisability), observability (detectability), invariant zeros, decoupling zeros, infinite zero structure, effect of feedback on zero structure, squaring down, diagonal and triangular decoupling .... etc can be directly displayed in terms of the special coordinate basis. Moreover, connections between the special coordinate basis and the important invariant subspaces of geometric theory of linear systems are established.

Periodic structures have attracted much attention in the past decades due to numerous applications such as metamaterials, antenna arrays, frequency selective surface (FSS), photonic band gap (PBG) or electromagnetic band gap (EBG) structures, etc. Infinite periodic structures are impossible in a realistic fabrication. When truncating infinite periodic structures, discontinuities occur at the interface and surface waves will be excited according to mode conversion, which leads to the edge effect of finite periodic structures. The prediction of wave natures in the edge cells of finite periodic structures is an indispensable procedure during the design of finite periodic structures. The conventional element-by-element method is rigorous and suitable for small finite arrays. However, it requires heavy computational resources for very large arrays. The well-established infinite periodic approximation approach neglecting the edge effect, although reduces the computational domain to a unit cell, is not sufficiently accurate for engineering applications.