矩阵方程AX=B的双反对称最佳逼近解

本文主要讨论下而两个问题并得到相关结果:问题Ⅰ:给定A ∈ R~(k×n),B ∈ R~(k×n),求X ∈ BASR~(n×n),使得AX=B.问题Ⅱ:给定X* ∈R~(n×n),求X使得‖X-X~*‖=minX∈S_E‖X-X~*‖,其中S_E是问题Ⅰ的解集合,‖·‖是Frobenius范数.通过对上述问题的讨论给出了问题Ⅰ解存在的充分必要条件和其解的一般表达式同时给出了问题Ⅱ的解,算法,和数值例子.     

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S={x₁,…,x_n} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (x_i,x_j)^e of the greatest common divisor of x_i and x_j is called the eth power GCD matrix on S, denoted by (S^e). Similarly we can define the eth power LCM matrix [S^e]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case max_x∈S{|G_S(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S^e)|[S^e]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.     

一般矩阵函数的不等式

本文研究一般矩阵函数不等式.运用一般矩阵函数的两个公式和lp-范数的性质,结出了两个矩阵运算的一般矩阵函数不等式.     

SIMS of organic layers with unknown matrix parameters: Locating the interface in dual beam argon gas cluster depth profiles

Four simple methods are evaluated to determine their accuracies for establishing the interface location in secondary ion mass spectrometry intensity depth profiles of organic layers where matrix effects have not been measured. Accurate location requires the separate measurement of each ion's matrix factor. This is often not possible, and so estimates using matrix-less methods are required. Six pure organic material interfaces are measured using many secondary ions to compare their locations from the four methods with those from full evaluation with matrix terms. For different secondary ions, matrix effects cause the apparent interface positions to vary over 20 nm. The shifts in the intensity profiles on going from a layer of P into a layer of Q are in the opposite direction to that for going from Q into P, so doubling layer thickness errors. The four methods are as follows: M1, use of the median interface position in the intensity profiles for the five lightest ions for 15 ≤ m/z ≤ 150; M2, extrapolation of the position for each ion to m/z = 0 for ions with m/z ≤ 150; M3, as M2 but for m/z ≤ 300; and M4, the extreme positions for all m/z ≤ 100. Comparison with the location using matrix terms shows their ranking, from best to worst, to be M4, M3, M1, and M2 with average errors of 10%, 12%, 14%, and 17%, respectively, of the profile interface full widths at half maximum. Use of pseudo-molecular ions is very much poorer, exceeding 50%, and should be avoided.     

Effect of matrix polymerization conditions on the structure and adsorption properties of porous polymers on the basis of divinylbenzene,acrylonitrile, and methyl methacrylate

A comparative study was carried out on the matrix polymerization of divinylbenzene, acrylonitrile, and methyl methacrylate in SBA-15 silica mesoporous molecular sieves. The occupancy of the matrix mesopores by the starting monomer, the medium, the polymerization temperature and time as well as the means of removing the exotemplate were all found to affect the spatial organization and porous structure of the polymer materials. Surplus occupancy of the mesopores by the monomer (1–1.5), polymerization in vacuum, and an alkaline method for matrix removal were found to be optimal. IR spectroscopy was used to find the conversion of the starting polymer by following the relative change in intensity of the vinyl group bands. Translated from Teoreticheskaya i éksperimental'naya Khimiya, Vol. 44, No. 6, pp. 371–375, November–December, 2008.     

The flow-through silica as the matrix to immobilize gold nanoparticles for HPLC applications

In the present study, the flow-through silica, featured with hierarchical pores, i.e., tunable mesopores and penetrable macropores, was attempted as the chromatographic stationary phase matrix to immobilize gold nanoparticles (AuNPs). It was first modified by mercapto groups (named as SiO₂-SH), and then by AuNPs (named as SiO₂-S-Au). Thanks to the characteristic macropores, the column backpressure of SiO₂-S-Au was comparable to SiO₂-SH, which effectively overcame the difficulty of high column backpressure upon the nanoparticles were introduced to the chromatographic matrix. Both the reversed-phase and hydrophilic interaction liquid chromatographic performance were observed on these two columns but with different selectivities. Hydrophobic, hydrophilic, hydrogen bond and electrostatic interactions between the SiO₂-S-Au stationary phase and analytes could contribute to the retention. The SiO₂-S-Au column showed excellent aqueous compatibility by “Stop-flow” test with the relative standard deviations (RSD) of analyte’s k (capacity factor) values from 0.59% to 2.88%. The reproducibility of SiO₂-S-Au was acceptable with RSDs of analyte’s k values in the range of 3.13%-5.03%. In addition, compared with the SiO₂-SH column, the SiO₂-S-Au column had better separation performance and selectivity. The results demonstrated that the flow-through silica was a promising matrix for nanoparticles with low backpressure and different selectivities.     

Bonding of multiple noble-gas atoms to CUO in solid neon: CUO(Ng)n (Ng=Ar, Kr, Xe; n=1, 2, 3, 4) complexes and the singlet-triplet crossover point

Laser-ablated U atoms co-deposited with CO in excess neon produce the novel CUO molecule, which forms distinct Ng complexes (Ng=Ar, Kr, Xe) with the heavier noble gases. The CUO(Ng) complexes are identified through CO isotopic and Ng reagent substitution and comparison to results of DFT frequency calculations. The U[bond]C and U[bond]O stretching frequencies of CUO(Ng) complexes are slightly red-shifted from neon matrix (1)Sigma(+) CUO values, which indicates a (1)A' ground state for the CUO(Ng) complexes. The CUO(Ng)(2) complexes in excess neon are likewise singlet molecules. However, the CUO(Ng)(3) and CUO(Ng)(4) complexes exhibit very different stretching frequencies and isotopic behaviors that are similar to those of CUO(Ar)(n) in a pure argon matrix, which has a (3)A" ground state based on DFT vibrational frequency calculations. This work suggests a coordination sphere model in which CUO in solid neon is initially solvated by four or more Ne atoms. Up to four heavier Ng atoms successively displace the Ne atoms leading ultimately to CUO(Ng)(4) complexes. The major changes in the CUO stretching frequencies from CUO(Ng)(2) to CUO(Ng)(3) provides evidence for the crossover from a singlet ground state to a triplet ground state.     

Activation and Coordination of Ammonia at [Cp*Ir(H)2]: NMR and Matrix Isolation Studies

¹H NMR exchange spectroscopy of a reaction mixture of [Cp*Ir(H)₄] ( 1 ; Cp*=1,2,3,4,5‐pentamethylcyclopentadienyl) and ammonia suggests an exchange of hydrogen atoms between the hydrido ligands and ammonia. Treatment of 1 with ND₃ led to an H/D exchange between ND₃ and the hydrido ligands of 1 . Subsequent studies showed that photolysis of 1 isolated in frozen argon matrices leads to the formation of the iridium compounds [Cp*Ir(H)₂] ( 2 ) and [Cp*Ir(H)₃] ( 4 ), as it was confirmed by IR spectroscopy. In the presence of water the aqua complex [Cp*Ir(H)₂(OH₂)] ( 3 ) was generated simultaneously. Accordingly, photolysis of 1 in an argon matrix doped with ammonia gave rise to the ammine complex [Cp*Ir(H)₂(NH₃)] ( 5 ). IR assignments were supported by calculations of the gas‐phase IR spectra of 1 – 5 by DFT methods.     

Matrix‐assisted laser ablation production of gold cluster ions from Au‐coated photonic crystals

A new strategy was explored to generate pure gold cluster ions, Au_n^+/?, from gold films deposited on solid substrates via a matrix‐assisted laser ablation technique. The gold films deposited on SiO₂‐particle‐assembled photonic crystals were demonstrated to be the most ideal compared with the films deposited on various glass slides. Dropped with a matrix of 2‐(4‐hydroxyphenylazo) benzoic acid and bombarded by nitrogen pulse laser (355 nm), they could release a series of Au_n⁺ with n more than 110 or Au_n^? with n more than 60 according to the data obtained by inline time‐of‐flight mass spectrometry. The gold‐deposited photonic crystal substrates could be stored at room temperature for at least 6 months. The method is hence steady and convenient in use. Copyright © 2012 John Wiley & Sons, Ltd.     

Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.