Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

In-gel deglycosylation of sodiumdodecyl sulfate polyacrylamide gel electrophoresis-separated glycoproteins for carbohydrate estimation by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry

Mass determination by mass spectrometric methods (electrospray ionization mass spectrometry (ESI-MS), matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOFMS)) of sodiumdodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE)-separated proteins is a well known procedure and reliable protocols are available. In our efforts to use the established methods to determine the molecular mass of the disulfide bridged, heterodimeric glycoprotein GP3 and to determine the carbohydrate content of each protein subunit we developed an in-gel chemical deglycosylation method. For this purpose we established experimental conditions that allow maximum extraction of the high molecular mass protein subunits and developed a routine method to apply the HF-pyridine deglycosylation protocol to proteins isolated from polyacrylamide gel pieces. The novel protocol and extraction procedure described can be used to analyze O-glycosylated proteins up to 150 kDa after SDS-PAGE separation.     

Survey of ligand field parameters of strong field d complexes obtained from the g matrix

Using a newly proposed approach involving an internally consistent set of equations, the ligand field parameters Δ/ξ, V/ξ and k are obtained from literature values of the g matrix for strong field d⁵ systems of various conformations in which |Δ/ξ|≤10. Qualitative analysis of the observed results is done using the Angular Overlap Model, AOM.     

The Effects of Charged Amino Acid Side-Chain Length on Diagonal Cross-Strand Interactions between Carboxylate- and Ammonium-Containing Residues in a β-Hairpin

The β-sheet is one of the common protein secondary structures, and the aberrant aggregation of β-sheets is implicated in various neurodegenerative diseases. Cross-strand interactions are an important determinant of β-sheet stability. Accordingly, both diagonal and lateral cross-strand interactions have been studied. Surprisingly, diagonal cross-strand ion-pairing interactions have yet to be investigated. Herein, we present a systematic study on the effects of charged amino acid side-chain length on a diagonal ion-pairing interaction between carboxylate- and ammonium-containing residues in a β-hairpin. To this end, 2D-NMR was used to investigate the conformation of the peptides. The fraction folded population and the folding free energy were derived from the chemical shift data. The fraction folded population for these peptides with potential diagonal ion pairs was mostly lower compared to the corresponding peptide with a potential lateral ion pair. The diagonal ion-pairing interaction energy was derived using double mutant cycle analysis. The Asp2-Dab9 (Asp: one methylene; Dab: two methylenes) interaction was the most stabilizing (−0.79 ± 0.14 kcal/mol), most likely representing an optimal balance between the entropic penalty to enable the ion-pairing interaction and the number of side-chain conformations that can accommodate the interaction. These results should be useful for designing β-sheet containing molecular entities for various applications.     

单A直和子空间

引入单直和子空间的概念,证明了线性空间可唯一分解为单直和子空间的直和,并且给出了计算方法.     

Photo-Induced Reactions between Glyoxal and Hydroxylamine in Cryogenic Matrices

In this paper, the photochemistry of glyoxal–hydroxylamine (Gly–HA) complexes is studied using FTIR matrix isolation spectroscopy and ab initio calculations. The irradiation of the Gly–HA complexes with the filtered output of a mercury lamp (λ > 370 nm) leads to their photoconversion to hydroxyketene–hydroxylamine complexes and the formation of hydroxy(hydroxyamino)acetaldehyde with a hemiaminal structure. The first product is the result of a double hydrogen exchange reaction between the aldehyde group of Gly and the amino or hydroxyl group of HA. The second product is formed as a result of the addition of the nitrogen atom of HA to the carbon atom of one aldehyde group of Gly, followed by the migration of the hydrogen atom from the amino group of hydroxylamine to the oxygen atom of the carbonyl group of glyoxal. The identification of the products is confirmed by deuterium substitution and by MP2 calculations of the structures and vibrational spectra of the identified species.     

适用于圆筒形增益区的同轴四镜腔的失调分析

采用矩阵光学方法,对适用于可见光的圆筒形增益区同轴四镜腔中的腔镜因倾斜而产生的失调特性进行了分析,给出了相关计算公式。该四镜腔由同轴的一个复曲面全反镜和两个光学玻璃加工的透镜和一个平面输出镜组成。以He-Ne激光器为例,给出了数值计算结果,关键元件复曲面镜的失调对输出的影响较大。这为腔元件的加工以及腔镜的安装调试提供了参考。     

New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy

A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.     

Secure Direct Communication Using Ensembles with the Same Compressed Density Matrix

In this paper, a quantum secure direct communication protocol using ensembles with the same density matrix is proposed. The two communication parties can realize the message transmission using this method through a quantum channel, each bit of information can be transmitted using an ensemble and read out through global measurement.The eavesdropping behavior can be detected through the channel diagnoses.