Local reconstruction of low‐rank matrices and subspaces

We study the problem of reconstructing a low‐rank matrix, where the input is an n × m matrix M over a field and the goal is to reconstruct a (near‐optimal) matrix that is low‐rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ‐close to a matrix of rank (together with d and ), this algorithm computes with high probability a rank‐d matrix that is ‐close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry by reading only the data structure and 2d additional entries of M. We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and , we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see: http://eccc.hpi-web.de/report/2015/128/ © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 607–630, 2017     

On a difference equation of the second order with an exponential coefficient

This paper is devoted to a generalization of some previous results, as we completely solve a linear homogeneous difference equation of the second order with an exponential coefficient.     

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.     

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.     

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.     

Evaluation of a novel metal–organic framework as an adsorbent for the extraction of multiclass pesticides from coconut palm (Cocos nucifera L.): An analytical approach using matrix solid‐phase dispersion and liquid chromatography

We report the synthesis, characterization, and application of [Zn(1,4‐benzenedicarboxylate)(H₂O)₂]_n , Zn(1,4‐benzenedicarboxylate)_0.99(NH₂‐1,4‐benzenedicarboxylate)_0.01(H₂O)₂]_n , [Zn(1,4‐benzenedicarboxylate)_0.95(NH₂‐1,4‐benzenedicarboxylate)_0.05(H₂O)₂]_n , and [Zn(1,4‐benzenedicarboxylate)_0.9(NH₂‐1,4‐benzenedicarboxylate)_0.1(H₂O)₂]_n as sorbents for the extraction of multiclass pesticides from coconut palm. Liquid chromatography with ultraviolet diode array detection was used as the analysis technique, and the experiments were performed at one fortification level (0.1 μg/g). The recoveries were 47–67, 51–70, 58–72, and 64–76% for [Zn(1,4‐benzenedicarboxylate)(H₂O)₂]_n , Zn(1,4‐benzenedicarboxylate)_0.99(NH₂‐1,4‐benzenedicarboxylate)_0.01(H₂O)₂]_n , [Zn(1,4‐benzenedicarboxylate)_0.95(NH₂‐1,4‐benzenedicarboxylate)_0.05(H₂O)₂]_n , and [Zn(1,4‐benzenelate)_0.95(NH₂‐1,4‐benzenedicarboxylate)_0.05(H₂O)₂]_n , and [Zn(1,4‐benzenedicarboxylate)_0.9(NH₂‐1,4‐benzenedicarboxylate)_0.1(H₂O)₂]_n , respectively, with relative standard deviation ranging from 1 to 7% (n = 3). Detection and quantification limits were 0.01–0.05 and 0.05–0.2 μg/g, respectively, for the different pesticides studied. The method developed was linear over the range tested (0.01–10.0 μg/g) with r ² > 0.9991. A direct comparison of [Zn(1,4‐benzenedicarboxylate)_0.9(NH₂‐1,4‐benzenedicarboxylate)_0.1(H₂O)₂]_n with the commercially available neutral alumina showed that [Zn(1,4‐benzenedicarboxylate)_0.9(NH₂‐1,4‐benzenedicarboxylate)_0.1(H₂O)₂]_n was a similar extracting phase for the pesticides investigated.     

Controlled Release of Hepatocyte Growth Factor from MPEG‐b‐(PCL‐ran‐PLLA) Diblock Copolymer for Improved Vocal Fold Regeneration

An in situ‐forming gel system comprised of diblock copolymer formed from polyethylene glycol (PEG) and polycaprolactone (PCL) {MPEG‐b‐(PCL‐ran‐PLLA)} could be used in controlled drug delivery for tissue remodeling. The purpose of this study is to demonstrate favorable vocal folds (VF) regeneration by using MPEG‐b‐(PCL‐ran‐PLLA) diblock copolymers (C97L3; CL/LA ratio 97:3) incorporating hepatocyte growth factor (HGF). Gradual release of HGF from C97L3 is detected and biochemical properties of released HGF are maintained. A scar is made with microscissors on both VFs in 32 rabbits, followed by injection of HGF‐only, C97L3‐only, or HGF‐C97L3 composite gel in the left side VF, while the right side VF is left untreated. In vivo fluorescence live imaging system demonstrates that C97L3 enables the sustained release of injected HGF in the scarred VF for 12 weeks. The histological analysis shows increased glycosaminoglycan including hyaluronic acid accumulation and decreased collagen deposition. Videokymographic analysis shows more favorable vibrations of HGF‐C97L3 treated VF mucosa, compared to other treatment groups. In conclusion, the controlled HGF release helps to regulate extracellular matrix synthesis, and leads to the eventual functional improvement of the scarred VF.

     

Computation of waveguide scattering matrix near thresholds

A waveguide occupies infinite strip with one or several narrows on a two-dimensional (2D) plane and is governed by the Helmholtz equation with Dirichlet boundary condition. On the waveguide continuous spectrum, which coincides with a half-axis, a scattering matrix is defined. At each point of the continuous spectrum this matrix has finite size, which changes at thresholds. The thresholds form a sequence of positive numbers increasing to infinity. Approximate calculation of the scattering matrix in a threshold vicinity requires special treatment. We discuss and compare two methods of numerical approximation to the scattering matrix near a threshold.