An analysis on the efficiency of Euler's method for computing the matrix pth root

It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton's method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler's method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior.     

Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X\in {\mathbb{C}}^{n\times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j{a}_j^{?}$ for some measurement vectors $a_1,\dots ,a_m$, i.e., the measurements are given by $b_j=\mathrm{tr}(X{a}_j{a}_j^{?})$. The case where the matrix $X=x{x}^{?}$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements $b_j=|〈x,a_j〉{|}^2$) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors $a_j$, $j=1,\dots ,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective t-design with $t=4$. In the Gaussian case, we require $m\ge Crn$ measurements, while in the case of 4-designs we need $m\ge \mathit{Cr}n\log ?(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.     

Phase II monitoring of changes in mean from high‐dimensional data

The generalized T² chart (GT‐chart), which is composed of the T² statistic based on a small number of principal components and the remaining components, is a popular alternative to the traditional Hotelling's T² control chart. However, the application of the GT‐chart to high‐dimensional data, which are now ubiquitous, encounters difficulties from high dimensionality similar to other multivariate procedures. The sample principal components and their eigenvalues do not consistently estimate the population values, and the GT‐chart relying on them is also inconsistent in estimating the control limits. In this paper, we investigate the effects of high dimensionality on the GT‐chart and then propose a corrected GT‐chart using the recent results of random matrix theory for the spiked covariance model. We numerically show that the corrected GT‐chart exhibits superior performance compared to the existing methods, including the GT‐chart and Hotelling's T² control chart, under various high‐dimensional cases. Finally, we apply the proposed corrected GT‐chart to monitor chemical processes introduced in the literature.     

Norm equalities and inequalities for three circulant operator matrices

In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered.     

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A₂ (resp. A₁⁽¹⁾. We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.     

Miniaturized matrix solid-phase dispersion and solid-phase clear-up combined with capillary electrophoresis for efficient determination of trace bioactive components in complicated sample matrix: Take Wubi Shanyao Pill as an example

Accurate quantitative analysis of trace analytes in a complicated matrix is a challenge in modern analytical chemistry. An appropriate analytical method is considered to be one of the most common gaps during the whole process. In this study, a green and efficient strategy based on miniaturized matrix solid-phase dispersion and solid-phase extraction combined with capillary electrophoresis was first proposed for extracting, purifying and determining target analytes from complicated matrix, using Wubi Shanyao Pill as an example. In detail, 60 mg of samples were dispersed on MCM-48 to obtain high yields of analytes, then the extract was purified with a solid-phase extraction cartridge. Finally, four analytes in the purified sample solution were determined by capillary electrophoresis. The parameters affecting the extraction efficiency of matrix solid-phase dispersion, purification efficiency of solid-phase extraction and separation effect of capillary electrophoresis were investigated. Under the optimized conditions, all analytes demonstrated satisfactory linearity (R²>0.9983). What's more, the superior green potential of the developed method for the determination of complex samples was confirmed by the Analytical GREEnness Metric Approach. The established method was successfully applied in the accurate determination of target analytes in Wubi Shanyao Pill and thus provided reliable, sensitive, and efficient strategy support for its quality control.     

Matrix solid-phase dispersion combined with micro-fractionation bioactivity evaluation screening polymethoxylated flavones from Citrus peels

Polymethoxyflavones were a unique class of natural and safe flavonoids containing two or more methoxy groups, which were also the most abundant edible part in Citrus peel. The optimum condition in the process of selective extraction of polymethoxylated flavones from Citrus peel by matrix solid-phase dispersion (MSPD) was as follows: SBA-15 as adsorbent, ethyl acetate as eluent, the mass ratio of adsorbent to sample 1:1, and the mixture of sample and adsorbent was ground for 3 min. Twelve antioxidants were successfully screened by micro-fractionation bioactivity evaluation assay, in which four of them were flavonoid glycosides, seven of them were polymethoxylated flavones, and one was phenylpropanoid. 1-sinapoly-β-D-glucopyranoside (1) was reported for the first time in Citrus peel. And antioxidant capacity of 1-sinapoly-β-D-glucopyranoside, 5, 7, 8, 3′, 4′, 5′-hexamethoxyflavone (6), hexamethoxyflavone (11), and 5, 6, 7, 4′-tetramethoxyflavone (7) were reported for the first time. Nobiletin (compound 8), 3, 5, 6, 7, 8, 3′, 4′-heptamethoxyflavone (9) and tangeretin (10) were isolated and purified by countercurrent chromatography combined with preparative liquid chromatography. Antioxidant activity evaluation indicated that the three isolated polymethoxylated flavones owned similar antioxidant activity. This study indicated that MSPD combined with micro-fractionation bioactive evaluation was efficient in screening bioactive compounds for rapid extraction and effective pinpointing bioactive substances in natural products.     

Exploring artificial neural networks to model interatomic and intermolecular potential energy surfaces

We demonstrate how a back-propagation artificial neural network can be trained to represent a potential energy surface (PES) in a formless manner with limited data points and exploited to predict interaction energies for configurations not included in the training set. A similar exercise is undertaken for predicting the eigenvalues and eigenvectors of a model Hamiltonian matrix that delicately depends on parameters and exhibits crossing of eigen values.     

Circulating Levels of Dephosphorylated-Uncarboxylated Matrix Gla Protein in Patients with Acute Coronary Syndrome

Vascular calcification contributes to the pathogenesis of coronary artery disease while matrix Gla protein (MGP) was recently identified as a potent inhibitor of vascular calcification. MGP fractions, such as dephosphorylated-uncarboxylated MGP (dp-ucMGP), lack post-translational modifications and are less efficient in vascular calcification inhibition. We sought to compare dp-ucMGP levels between patients with acute coronary syndrome (ACS), stratified by ST-elevation myocardial infarction (STEMI) and non-ST-elevation myocardial infarction (NSTEMI) status. Physical examination and clinical data, along with plasma dp-ucMGP levels, were obtained from 90 consecutive ACS patients. We observed that levels of dp-ucMGP were significantly higher in patients with NSTEMI compared to STEMI patients (1063.4 ± 518.6 vs. 742.7 ± 166.6 pmol/L, p < 0.001). NSTEMI status and positive family history of cardiovascular diseases were only independent predictors of the highest tertile of dp-ucMGP levels. Among those with NSTEMI, patients at a high risk of in-hospital mortality (adjudicated by GRACE score) had significantly higher levels of dp-ucMGP compared to non-high-risk patients (1417.8 ± 956.8 vs. 984.6 ± 335.0 pmol/L, p = 0.030). Altogether, our findings suggest that higher dp-ucMGP levels likely reflect higher calcification burden in ACS patients and might aid in the identification of NSTEMI patients at increased risk of in-hospital mortality. Furthermore, observed dp-ucMGP levels might reflect differences in atherosclerotic plaque pathobiology between patients with STEMI and NSTEMI.