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     

Analysis of metabolites of glucose pathways in human erythrocytes by analytical isotachophoresis

A method for the quantitative determination of the major anionic constituents of fountain solutions, typically mono-, di- and hydroxycarboxylates, alkylbenzenesulfonates, and inorganic anions, including orthophosphate and polyphosphates, is presented here for the first time. The analytical problems arising from extensive co-elution of many of these analytes on an ion-exchange column have been resolved through a combination of (i) careful selection of the concentration gradient of the sodium hydroxide eluent; (ii) parallel analysis by ion-exclusion chromatography; and (iii) determination of total phosphorus by inductively coupled plasma atomic emission spectrometry.     

Isotachophoretic analysis of the dihydrofolate reductase reaction in the presence of methotrexate and ascorbic acid

The antifolate methotrexate (MTX) is widely used in cancer chemotherapy. In this study, we show that MTX (MTX-Glu1) and MTX-polyglutamates (MTX-Glu2-5) strongly inhibited the growth of the leukemic cell line MOLT-4. This effect, however, was mitigated by ascorbic acid. We investigated whether ascorbic acid is able to reduce dihydrofolic acid (DHF) to tetrahydrofolic acid (THF) directly or by circumventing the MTX inhibition of dihydrofolate reductase (DHFR). The inhibition of this NADPH-dependent reduction of DHF by MTX-Glun in the absence or presence of ascorbate, was determined by analytical isotachophoresis. Using 0.01 M HCl/histidine, pH 6.0, as a leading electrolyte (L) and 0.005 M 2-(N-morpholino)ethanesulfonic acid (MES)/histidine, pH 6.0, as a terminating electrolyte (T), MTX-Glun derivatives including MTX-Glu1 could be easily separated, whereas the quantitative estimation of THF was not possible. A quantitative characterization of the DHFR reaction by measuring NADPH, NADP+ and ascorbate was achieved with another system (L: 0.01 M HCI/beta-alanine, pH 3.73; T: 0.01 M caproic acid, pH 3.27). Nanomolar concentrations of MTX-Glu1-5 inhibited consumption of NADPH and production of NADP+. Ascorbic acid was not able to reduce DHF, neither directly nor after inhibition of DHFR by MTX. However, ascorbic acid seemed to diminish the oxidation of THF and this may account for its capacity to reduce the inhibitory effect of MTX on MOLT-4 cells.     

Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

Old and new from SOR

From the long history of Successive Over Relaxation (SOR) between the end of second world war and today three points are considered: (1) Classical results of Young and Varga are described. (2) It is shown how results on semiiterative methods can be used to derive these classical results in a unifying way and to compare SOR with other iterative methods. (3) In the last 15 years the application of SOR to compute the stationary distribution of a homogeneous Markov chain has been discussed. These results are reported, considering especially the term extended convergence introduced by Kontovasilis, Plemmons and Stewart.     

Mikrochemie

Ohne Zusammenfassung     

Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel

We consider mass-conserving self-similar solutions for Smoluchowski’s coagulation equation with kernel K(ξ,η)=(ξη) ^λ with λ∈(0,1/2). It is known that such self-similar solutions g(x) satisfy that x ^?1+2λ g(x) is bounded above and below as x→0. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function h(x)=h _λ x ^?1+2λ g(x) in the limit λ→0. It turns out that \(h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)\) as x→0. As x becomes larger h develops peaks of height 1/λ that are separated by large regions where h is small. Finally, h converges to zero exponentially fast as x→∞. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.