An atlas for tridiagonal isospectral manifolds

Let ${\cal T}_\Lambda$ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix $\Lambda$ with simple spectrum. We introduce {\it bidiagonal coordinates}, charts defined on open dense domains forming an explicit atlas for ${\cal T}_\Lambda$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\cal T}_\Lambda$ now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.Comment: Fixed typos; 16 pages, 3 figure

Stereospecific reaction of sulfonimidoyl fluorides with Grignard reagents for the synthesis of enantioenriched sulfoximines

Sulfonimidoyl halides have previously shown poor stability and selectivity in reaction with organometallic reagents. Here we report the preparation of enantioenriched sulfonimidoyl fluorides and their stereospecific reaction at sulfur with Grignard reagents. Notably the first enantioenriched alkyl sulfonimidoyl fluorides are prepared, including methyl. The nature of the N-group is important to the success of the stereocontrolled sequence to sulfoximines

Simulator based human performance assessment in a ship engine room using functional near-infrared spectroscopy

80% of accidents that occur in the maritime sector are due to human error. These errors could be the result of seafarer training coupled with a high mental workload due to the addition of various working conditions. The aim of this study is to evaluate the effect of various stressors on human performance on engine room operations. To achieve this aim, a simulator study was conducted to investigate the influence of training and working conditions on human performance for the purposes of fault detection and correction in a maritime engine room. 20 participants were recruited for each investigation of performance shaping factors (PSF); for the first test, half received practical training with the engine room software interface, while the other half were provided with paper-based instructions. The remaining tests were conducted with all participants equally practically trained. The participants interacted with a TRANSAS technological simulator series 5000. This is a 1:1 simulation of a ship engine room. The participants took part in a 30-minute scenario where they had to detect and correct a fault with the ballasting system. During this interaction, half of the participants experienced simulated, adverse performance shaping factors, which were distraction, fatigue and an increased workload. The other half were given a standard task. Functional near-infrared spectroscopy (fNIRS) was utilised to measure neurophysiological activation from the dorsolateral prefrontal cortex (DLPFC). The results indicated increased activation of lateral regions of the DLPFC during fault correction, this trend was enhanced due to PSF’s and training, i.e. participants who received paper-based instructions showed greater activation when conducting the standard task and had an exponential increase in activation when dealing with the addition of an adverse PSF. The results are discussed with respect to the neural efficiency of the operator during high mental workload. From the results of this study a scientific human error model was developed and can be used by the maritime industry to better evaluate and understand human error causation and the effect of PSF on seafarers. The impact of this study could reduce the frequency of occurrence of human error, reduce the financial impact that human error has on the maritime sector and reduce injuries and fatalities

Construction of KP Hierarchies in Terms of Finite Number of Fields and their Abelianization

The $2M$-boson representations of KP hierarchy are constructed in terms of $M$ mutually independent two-boson KP representations for arbitrary number $M$. Our construction establishes the multi-boson representations of KP hierarchy as consistent Poisson reductions of standard KP hierarchy within the $R$-matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of \Win1\, algebra made out of arbitrary even number of boson fields.Comment: 12 p., LaTeX, minor typos corrected, BGU-93/2/June-P

Dynamics of the symmetric eigenvalue problem with shift strategies

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_\sigma$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal T}_\Lambda$, the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix $\Lambda$. More precisely, a (generic) shift s \in \RR defines a map $F_s: {\cal T}_\Lambda \to {\cal T}_\Lambda$. A strategy \sigma: {\cal T}_\Lambda \to \RR specifies the shift to be applied at $T$ so that $F_\sigma(T) = F_{\sigma(T)}(T)$. Good shift strategies should lead to fast deflation: some off-diagonal coordinate tends to zero, allowing for reducing of the problem to submatrices. For topological reasons, continuous shift strategies do not obtain fast deflation; many standard strategies are indeed discontinuous. Practical implementation only gives rise systematically to bottom deflation, convergence to zero of the lowest off-diagonal entry $b(T)$. For most shift strategies, convergence to zero of $b(T)$ is cubic, $|b(F_\sigma(T))| = \Theta(|b(T)|^k)$ for $k = 3$. The existence of arithmetic progressions in the spectrum of $T$ sometimes implies instead quadratic convergence, $k = 2$. The complete integrability of the Toda lattice and the dynamics at non-smooth points are central to our discussion. The text does not assume knowledge of numerical linear algebra.Comment: 22 pages, 4 figures. This preprint borrows heavily from the unpublished preprint arXiv:0912.3376 but is adapted for a different audienc

Explicit Integration of the Full Symmetric Toda Hierarchy and the Sorting Property

We give an explicit formula for the solution to the initial value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szeg\"{o}, and is also interpreted as a consequence of the QR factorization method of Symes \cite{symes}. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridiagonal formulae are given for the case of matrices with $2M+1$ nonzero diagonals.Comment: 13 pages, Latex