Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of $G$ are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on ${\mathbb{C}}$. It follows that all the Kogan-Zelevinsky integrable systems on $G$ have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct a class of integrable systems with complete Hamiltonian flows associated to {\it generalized Bruhat cells} which are defined using arbitrary sequences of elements in the Weyl group of $G$, and we obtain the results for double Bruhat cells through the so-called open {\it Fomin-Zelevinsky embeddings} of (reduced) double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide global coordinates on double Bruhat cells, called {\it Bott-Samelson coordinates}, in which all the Fomin-Zelevinsky minors become polynomials and the Poisson structure can be computed explicitly.Comment: Title slightly changed; Section 1.3 expanded; some typos correcte

Assessment of carotid atherosclerotic disease using three-dimensional cardiovascular magnetic resonance vessel wall imaging: comparison with digital subtraction angiography.

BACKGROUND:A three-dimensional (3D) cardiovascular magnetic resonance (CMR) vessel wall imaging (VWI) technique based on 3D T1 weighted (T1w) Sampling Perfection with Application-optimized Contrast using different flip angle Evolutions (SPACE) has recently been used as a promising CMR imaging modality for evaluating extra-cranial and intra-cranial vessel walls. However, this technique is yet to be validated against the current diagnostic imaging standard. We therefore aimed to evaluate the diagnostic performance of 3D CMR VWI in characterizing carotid disease using intra-arterial digital subtraction angiography (DSA) as a reference. METHODS:Consecutive patients with at least unilateral > 50% carotid stenosis on ultrasound were scheduled to undergo interventional therapy were invited to participate. The following metrics were measured using 3D CMR VWI and DSA: lumen diameter of the common carotid artery (CCA) and segments C1-C7, stenosis diameter, reference diameter, lesion length, stenosis degree, and ulceration. We assessed the diagnostic sensitivity, specificity, accuracy, and receiver operating characteristic (ROC) curve of 3D CMR VWI, and used Cohen's kappa, the intraclass correlation coefficient (ICC), and Bland-Altman analyses to assess the diagnostic agreement between 3D CMR VWI and DSA. RESULTS:The ICC (all ICCs ≥0.96) and Bland-Altman plots indicated excellent inter-reader agreement in all individual morphologic measurements by 3D CMR VWI. Excellent agreement in all individual morphologic measurements were also found between 3D CMR VWI and DSA. In addition, 3D CMR VWI had high sensitivity (98.4, 97.4, 80.0, 100.0%), specificity (100.0, 94.5, 99.1, 98.0%), and Cohen's kappa (0.99, 0.89, 0.84, 0.96) for detecting stenosis > 50%, stenosis > 70%, ulceration, and total occlusion, respectively, using DSA as the standard. The AUC of 3D CMR VWI for predicting stenosis > 50 and > 70% were 0.998 and 0.999, respectively. CONCLUSIONS:The 3D CMR VWI technique enables accurate diagnosis and luminal feature assessment of carotid artery atherosclerosis, suggesting that this imaging modality may be useful for routine imaging workups and provide comprehensive information for both the vessel wall and lumen

Simulation of microstructural evolution in directional solidification of Ti-45at.%Al alloy using cellular automaton method

The microstructural evolution of Ti-45 at.%Al alloy during directional solidification was simulated by applying a solute diffusion controlled solidification model. The obtained results have shown that under high thermal gradients the stable primary spacing can be adjusted via branching or competitive growth. For dendritic structures formed under a high thermal gradient, the secondary dendrite arms are developed not very well in many cases due to the branching mechanism under a constrained dendritic growth condition. Furthermore, it has been observed that, with increasing pulling velocity, there exists a cell/dendrite transition region consisting of cells and dendrites, which varies with the thermal gradient in a contradicting way, i.e. increase of the thermal gradient leading to the decrease of the range of the transition region. The simulations agree reasonably well with experiment results