On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

The many limits of mixed means

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a ₀,b ₀ we define

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Convolution quadrature and discretized operational calculus. I

Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals

Potentiometric studies on the complexation equilibria between some trivalent lanthanide metal ions and biologically active 2-hydroxy-1-naphthaldehyde thiosemicarbazone (HNATS)

Summary The chelation behaviour of some trivalent lanthanide and yttrium metal ion with biologically active 2-hydroxy-1-naphthaldehyde thiosemicarbazone (HNATS) has been investigated by potentiomotric measurements at 20±0.5°C in 75% (v/v) dioxane-water medium at various ionic strengths of sodium perchlorate. The method of Bjerrum and Calvin, as modified by Irving and Rossotti has been used to find out the values of (average number of ligand bound per metal ion) andpL (free ligand exponent). The formation constants of metal chelates have been computed on a PC-XT computer, using a program patterned after that of Sullivan et al. to give _n values using weighted least squares method. TheS _min values (S _min =²) have been calculated. The order of formation constants of chelates was found to be: La³⁺3+3+3+3+3+3+3+3+3+. The formation constants of the chelates formed have been correlated to size and ionization potentials of the metal ions.

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Potentiometrische Untersuchungen der Komplexierungsgleichgewichte zwischen einigen trivalenten Metallionen und biologisch aktivem 2-Hydroxy-1-naphthaldehyd-thiosemicarbazon (HNATS)

Zusammenfassung Es wurde das Chelierungsverhalten einiger trivalenter Lanthanidenionen mit biologisch aktivem 2-Hydroxy-1-naphthaldehyd-thiosemicarbazon (HNATS) mittels potentiometrischer Messungen bei 20±0.5°C in 75% (v/v) Dioxan-Wasser bei verschiedener Ionenstärke an Natriumperchlorat untersucht. Die Methode nach Bjerrum und Calvin in der Modifikation nach Irving und Rossotti wurde zur Ermittlung der Werte (mittlere Anzahl an Liganden pro Metallion) undpL (Exponent an freiem Liganden) verwendet. Die Komplexbildungskonstanten wurden in Anlehnung an Sullivan et al. an einem PC-XT Computer errechnet. DieS _min-Werte (S _min=²) wurden ebenfalls bestimmt. Die Reihung der Chelatbildungskonstanten war: La³⁺3+3+3+3+3+3+3+3+3+. Die Bildungskonstanten sind mit der Größe und dem Ionisierungspotential der Metallionen zu korrelieren.

     

How likely is Polya's drunkard to stay in x⩾ y⩾z?

In his celebrated paper, Polya has considered the random walk in the three-dimensional (cubic) lattice and showed that the probability of return to the origin is less than 1. Subsequent authors have shown that the probability is %34.053.... Here we consider the same random walk, with the restriction that the drunkard is only allowed to stay inxyz. It is shown that his probability of returning to the originand staying in the allowed region is %6.4844....     

ZnCl2/粘土-SA01催化剂上二苯甲烷的合成

在环境友好ZnCl2/粘土-SA01催化剂上合成了二苯甲烷,较系统地考察了负载量、苯/苄基氯摩尔比、催化剂用量、反应温度、反应时间对该傅克反应的影响,并与催化剂的表面性质关联.实验结果表明,该催化剂具有良好的催化活性和稳定性并易于回收重复使用.     

钢中Si、Mn的电子探针定量分析与标样

采用特制的Fe-Si、Fe-Mn合金系列标样，用电子探针测定了钢或铁基合金中低含量的Si和Mn元素、与用纯元素作标样、经ZAF修正的结果相比较，提高了分析准确度．     

在H－ZSM－5沸石上甲醇转化为汽油的初始产物分布

在Ｈ－ＺＳＭ－５沸石上甲醇转化为汽油的初始产物分布姜玄珍（浙江大学化学系，杭州３１００２７）Ｒ．Ｆ．Ｈｏｗｅ（新南威尔士大学物理化学系，澳大利亚）关键词Ｈ－ＺＳＭ－５沸石，甲醇转化汽油，产物分布以氢型ＺＳＭ－５沸石作催化剂转化甲醇为汽油（ＭＴＧ），在...     

配合物的生成热和配体的质子化热之间的直线焓关系——Ⅶ.Ni(Ⅱ)-2,2′-联吡啶-N-(间位取代苯基)氨基乙酸三元体系

有关Ni(Ⅱ)-N-(间位取代苯基)氨基乙酸二元体系的直线焓和直线熵关系,前文已有报道,本文报告Ni(Ⅱ)-2,2′-联吡啶-N-(间位取代苯基)氨基乙酸(Ni(Ⅱ)-biPy-m-RPhG:R=CH_3,H,CH_3O,Cl)三元体系的生成热研究,发现在此三元体系中亦存在良好的直线焓和直线熵关系。