Controllability completion problems of partial upper triangular matrices

The main result of this paper states sufficient conditions for the existence of a completion A_c of an n × n partial upper triangular matrix A, such that the pair (A_cB) has prescribed controllability indices, being B an n×m matrix. If A is a partial Hessenberg matrix some conditions may be dropped. An algorithm that obtains a completion A_c of A such that pair (A_ce_k) is completely controllable, where e_k is a unit vector, is used to proof the results.     

Extracting partial canonical structure for large scale eigenvalue problems

We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted on problems with known canonical structure and varying ill-conditioning are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two mathematical problems of canonical quantization. I

Power Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 87, No. 1, pp. 22–23, April, 1991.     

Correction of improper linear programming problems in canonical form by applying the minimax criterion

Given an inconsistent system of linear algebraic equations, necessary and sufficient conditions are established for the solvability of the problem of its matrix correction by applying the minimax criterion with the assumption that the solution is nonnegative. The form of the solution to the corrected system is presented. Two formulations of the problem are considered, specifically, the correction of both sides of the original system and correction with the right-hand-side vector being fixed. The minimax-criterion correction of an improper linear programming problem is reduced to a linear programming problem, which is solved numerically in MATLAB.     

On a certain canonical form

We amplify the well-known result due to Dlab and Ringel on the reduction of a real rectangular matrix to canonical form by formally complex transformations of rows and columns. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 514–519, April, 2000.     

Matrix pencils completion problems

In this paper we give a partial solution to the challenge problem posed by Loiseau et al. in [J. Loiseau, S. Mondié, I. Zaballa, P. Zagalak, Assigning the Kronecker invariants of a matrix pencil by row or column completion, Linear Algebra Appl. 278 (1998) 327-336], i.e. we assign the Kronecker invariants of a matrix pencil obtained by row or column completion. We have solved this problem over arbitrary fields.     

On the generalized Lorenz canonical form

This short note is to briefly introduce the new notion of generalized Lorenz canonical form (GLCF), which contains the classical Lorenz system and the newly discovered Chen system as two extreme cases, along with infinitely many chaotic systems in between. It also points out that some recently reported chaotic systems are special cases of the GLCF.     

On the generalized Lorenz canonical form

This short note is to briefly introduce the new notion of generalized Lorenz canonical form (GLCF), which contains the classical Lorenz system and the newly discovered Chen system as two extreme cases, along with infinitely many chaotic systems in between. It also points out that some recently reported chaotic systems are special cases of the GLCF.     

The completion problem for partial packings

Packings (resolutions) of designs have been of interest to combinatorialists in recent years as a way of creating new designs from old ones. Line packings of projective 3-space were the first packings studied, but it is still unknown when a partial packing can be completed to a packing in this case. In this paper we show that there is no guarantee from a combinatorial point of view of completing such a partial packing even when the deficiency is 2. In particular, we construct for every odd prime power q a set of 2(q ²+1) lines which doubly cover the points of PG(3,q) and yet cannot be partitioned into two spreads (resolution classes). The method is based on manipulations of primitive elements of finite fields.This work was supported in part by the National Research Council of Canada.     

On the completion of partial latin squares

In this paper a certain condition on partial latin squares is shown to be sufficient to guarantee that the partial square can be completed, namely, that it have fewer than n entries, and that at most $\text{[}\text{(n + 1)}\text{2}\text{]}$ of these lie off some line, where n is the order of the square. This is applied to establish that the Evans conjecture is true for n ? 8; i.e., that given a partial latin square of order n with fewer than n entries, n ? 8, the square can be completed. Finally, the results are viewed in a conjugate way to establish different conditions sufficient for the completion of a partial latin square.     

Matrix completion problems of block type

The solvability, either unconditional or under certain conditions, is proved for three matrix completion problems of block type. Being a problem of block type means that a matrixA must be constructed with a prescribed characteristic polynomial and one or several blocks in a 2 × 2 partition also prescribed. For the problems under analysis, one prescribes, respectively, (a) the blockA ₁₂; (b) the blockA ₁₁; (c) the blocksA ₁₁ andA ₁₂. We discuss our experience in implementing the algorithms that solve the problems above as procedures in the symbolic computation system MAPLE. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 863–873, June, 2000. The first author was supported by the Russian Foundation for Basic Research under grant No. 97-01-00927.     

A canonical form for relativistic dynamic equation

In this paper, an elementary proof of representation for Clifford AlgebraCℓ(3, 0) and some of its applications are given.     

On the canonical form of the electromagnetic field

Abstract Using the decomposition of an antisymmetric 2-tensor as a sum of two orthogonal bivectors, the various canonical forms of the electromagnetic tensor field are analyzed, recovering known results. However, introducing 1+3 spacetime splitting techniques, the canonical forms are associated to special frames and observers and this helps to clarify the role played by “measurable” quantities (electric and magnetic fields, Poynting vector) in the classification problem itself. Keywords: Electromagnetic field, Canonical form Mathematics Subject Classification (2000): 83A05, 78A25     

A Markovian canonical form of second-order matrix-exponential processes

Besides the fact that – by definition – matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that – in two dimensions – the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP(2)≡MAP(2)≡MEP(2)$\mathrm{AMAP}(2)\equiv \mathrm{MAP}(2)\equiv \mathrm{MEP}(2)$. For higher orders, these equivalences do not hold.     

The canonical order and optimization problems

Using the partial order technique, we describe a subclass of objective functions taking their optimum at the greedy point of a given feasible polyhedron inR ⁿ. Supported by the Belarusian Foundatmental Research Found