Backward Error Analysis of Factorization Algorithms for Symmetric and Symmetric Triadic Matrices

We consider the $LBL^T$ factorization of a symmetric matrix where $L$ is unit lower triangular and $B$ is block diagonal with diagonal blocks of order $1$ or $2$. This is a generalization of the Cholesky factorization, and pivoting is incorporated for stability. However, the reliability of the Bunch-Kaufman pivoting strategy and Bunch's pivoting method for symmetric tridiagonal matrices could be questioned, because they may result in unbounded $L$. In this paper, we give a condition under which $LBL^T$ factorization will run to completion in inexact arithmetic with inertia preserved. In addition, we present a new proof of the componentwise backward stability of the factorization using the inner product formulation, giving a slight improvement of the bounds in Higham's proofs, which relied on the outer product formulation and normwise analysis. We also analyze the stability of rank estimation of symmetric indefinite matrices by $LBL^T$ factorization incorporated with the Bunch-Parlett pivoting strategy, generalizing results of Higham for the symmetric semidefinite case. We call a matrix triadic if it has no more than two non-zero off-diagonal elements in any column. A symmetric tridiagonal matrix is a special case. In this paper, we display the improvement in stability bounds when the matrix is triadic

The Nature of Retrograde Analysis for Chinese Chess

Retrograde analysis has been successfully applied to solve Awari and construct 6-piece Western chess endgame databases. However, its application to Chinese chess is limited because of the special rules about indefinite move sequences. Problems caused by the most influential rule, checking indefinitely were successfully solved in practical cases, with $50$ selected endgame databases constructed in accord with this rule, where the 60-move-rule was ignored. Other special rules have much less impact on contaminating the databases, as verified by the rule-tolerant algorithms. For constructing complete endgame databases, we need rigorous algorithms. There are two rule sets in Chinese chess: Asian rule set and Chinese rule set. In this paper, an algorithm is successfully developed to construct endgame databases in accord with the Asian rule set. The graph-theoretical properties are also explored as well

Acute Myocardial Infarction Presenting with Sudden Death Owing to Cardiac Rupture

A 74-year-old woman was admitted to our hospital with a clinical diagnosis of angina pectoris. An electrocardiogram performed in the emergency room showed a sinus rhythm and no evidence of myocardial ischemia or infarction. On the 2nd day, the patient experienced persistent protracted chest pain that could not be relieved by sublingual nitroglycerine administration, which was subsequently followed by sudden hemodynamic collapse, loss of consciousness, and apparent electromechanical dissociation a few minutes later. Immediate cardiopulmonary resuscitation was performed. Instantaneous electrocardiography demonstrated acute extensive anterior and inferior myocardial infarction. Two-dimensional echocardiography was performed promptly, which revealed the presence of pericardial effusion and an echo-dense suspected blood clot within the pericardial cavity, as well as diffuse severe hypocontractility of the whole myocardium. Urgent surgical intervention with subxiphoid incision was performed at the bedside of the patient. During the surgery, a large quantity of blood was drained from the pericardial cavity. The patient died soon, prior to being sent to the operation room, because of protracted cardiac standstill. In conclusion, in a patient with acute myocardial infarction who presents with persistent, protracted chest pain and hemodynamic instability, cardiac rupture should be considered and echocardiography should be performed promptly

Kerr-effect-based quantum logical gates in decoherence-free subspace

Efficient implementations of two (or three) qubit logical gates are critical for the large-scale realization of quantum computation in decoherence-free subspace (DFS) immune to the influence of decoherence effect. In this paper, we propose some schemes for setting up a family of quantum control gates, including controlled-NOT (CNOT), Toffoli, and Fredkin gates for two or three logical qubits by means of cross-Kerr nonlinearities in DFS. These three logical gates require neither complicated quantum computational circuits nor auxiliary photons (or entangled states). The success probabilities of three logical gates are approximate unit by performing the corresponding classical feed-forward operations based on the different measuring results of the X homodyne detectors, and their fidelities are robust against the photon loss with the current technology.The proposed logical gates rely on only simple linear-optics elements, available single qubit operations, and mature measurement methods, making our proposed gates be feasible and efficient in practical applications.Comment: 11 pages, 9 figure

Homogeneous ACM bundles on exceptional isotropic Grassmannians

In this paper, we characterize homogeneous arithmetically Cohen-Macaulay (ACM) bundles over exceptional isotropic Grassmannians in terms of their associated data. We show that there are only finitely many irreducible homogeneous ACM bundles by twisting line bundles over exceptional isotropic Grassmannians. As a consequence, we prove that some exceptional isotropic Grassmannians are of wild representation type.Comment: 18 pages. arXiv admin note: text overlap with arXiv:2206.0917