Simultaneous Row Research Articles

Simultaneous disturbance rejection and regular row by row decoupling with stability: a geometric approach

The simultaneous disturbance rejection problem and regular row by row decoupling problem with stability is solved here through a geometric approach. It is shown in this paper that the combined problem has a solution if and only if each problem, separately, has a solution. >

Electrical stimulation-assisted rowing exercise in spinal cord injured people. A pilot study.

Recently a FES (functional electrical stimulation)-assisted rowing machine was developed to enhance cardiovascular training in people with spinal cord injuries. The machine was assessed in terms of its efficacy as a training tool. Six patients who were quadriplegic (C6-T1) and 2 who were paraplegic (T3-6) completed a series of three tests in succession: (1) leg stimulation only (quadriceps and hamstring groups)--'Stim', (2) arm row only--'Row' and (3) simultaneous row and stimulation--'R & S'. Measurements recorded included oxygen uptake (VO2), minute ventilation (Ve), respiratory exchange ratio (RER), heart rate (HR) and blood pressure (BP). In addition, 6 out of the 8 subjects took part in a qualitative assessment comprising a guided interview exploring the subject's perception of the machine and test. Significant increases in VO2 were demonstrated between the three tests with R & S producing mean steady-state values of 16.34 nm (+/- 0.74) ml/kg/min (83% of maximum). These values represented a 12% increase over Row alone. Of interest was the qualitative assessment which revealed that subjects perceived R & S to be easier than Row despite the higher levels of VO2 observed. The results suggest that the rowing machine represents a potentially valuable hybrid training device that may significantly reduce risk factors for cardiovascular disease and improve the quality of life of people with SCI.

Diagonalization in Jordan pairs

As every student of linear algebra knows, a rectangular matrix over a division ring can be diagonalized by elementary row and column operations. Similarly, there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations. Since matrices of this kind form the main examples of Jordan pairs, it is natural to ask whether similar results hold in the Jordan setting. We show that this is in fact the case, study the obstruction to diagonalizability, the defect, and also prove that nondegenerate Jordan pairs admit a rank function sharing many properties with classical matrix rank. Let V= (V+, V) be a Jordan pair and S = {e,, . . . . e,} a set of orthogonal idempotents. The S-diagonal elements are those in xi= 1 V,(e,). Now suppose that V is nondegenerate and satisfies dcc on principal inner ideals. An element x E V” (a = i) is called diagonalizabfe if x is S-diagonal for some S consisting of division idempotents. To see that, for matrices, this is the same as the usual notion of diagonalizability, suppose that V has in addition act on principal inner ideals. Then V contains a frame, that is, a finite set F of orthogonal division idempotents such that V,,(F) = 0. For rectangular or hermitian matrices over division rings, F can be taken to consist of the diagonal matrix units (e,? = Eii) whereas for alternating matrices over a field, e,? = Eli1,2iE,,,zi-, . The Jordan analogues of elementary row and column operations are the inner automorphisms B( V:(e), V;(e)) and /?( V:(e), V,(e)) (where e E F) which generate the group of F-elementary automorphisms of V [6]. Any set of orthogonal division idempotents can be transformed (up to association) into F by an F-elementary automorphism [6, Th. 21. It follows that x is diagonalizable if and only if cp +(x) is F-diagonal for some elementary automorphism cp of V. This shows the equivalence with the usual definition.

On the Probable Performance of Heuristics for Bandwidth Minimization

Most research in algorithm design relies on worst-case analysis for performance comparisons. Unfortunately, worst-case analysis does not always provide an adequate measure of an algorithm’s effectiveness. This is particularly true in the case of heuristic algorithms for hard combinatorial problems. In such cases, analysis of the probable performance can yield more meaningful results and can provide insight leading to better algorithms. The problem of minimizing the bandwidth of a sparse symmetric matrix by performing simultaneous row and column permutations, is an example of a problem for which there are well-known heuristics whose practical success has lacked a convincing analytical explanation. A class of heuristics introduced by Cuthill and McKee in 1969, and referred to here as the level algorithms, are the basis for bandwidth minimization routines that have been widely used for over a decade. At the same time, it is easy to construct examples, showing that the level algorithms can produce solutions t...

Complexity Results for Bandwidth Minimization

We present a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form (“bandwidth 2”), whenever it is possible to do so using simultaneous row and column permutations. On the other hand when an arbitrary integer k and graph G are given, we show that it is $NP$-complete to determine whether or not there exists an ordering of the vertices such that the adjacency matrix has bandwidth $ \leqq k$, even when G is restircted to the class of free trees with all vertices of degree $ \leqq 3$. Related problems for acyclic directed graphs (upper triangular matrices) are also discussed.

Matrices which can be optimally scaled

Sufficient conditions are given for a matrix to be optimally scalable in the sense of minimizing its condition number. In particular, in the case of simultaneous row- and column-scaling and subordinate to thel 1- orl ?-norm the minimal condition number is achieved for fully indecomposable matrices.