Bounds for the error of linear systems of equations using the theory of moments

Consider the system, of linear equations Ax = b where A is an n × n real symmetric, positive definite matrix and b is a known vector. Suppose we are given an approximation to x, ξ, and we wish to determine upper and lower bounds for ∥ x − ξ ∥ where ∥ ··· ∥ indicates the euclidean norm. Given the sequence of vectors {r_i}_{i^k = 0}, where r_i = Ar_{i − 1} and r₀ = b − Aξ, it is shown how to construct a sequence of upper and lower bounds for ∥ x − ξ ∥ using the theory of moments.     

Copper‐Containing SiCN Precursor Ceramics (Cu@SiCN) as Selective Hydrocarbon Oxidation Catalysts Using Air as an Oxidant

A molecular approach to metal‐containing ceramics and their application as selective heterogeneous oxidation catalysts is presented. The aminopyridinato copper complex [Cu₂(Ap^TMS)₂] (Ap^TMSH=(4‐methylpyridin‐2‐yl)trimethylsilanylamine) reacts with poly(organosilazanes) via aminopyridine elimination, as shown for the commercially available ceramic precursor HTT 1800. The reaction was studied by ¹H and ¹³C NMR spectroscopy. The liberation of the free, protonated ligand Ap^TMSH is indicative of the copper polycarbosilazane binding. Crosslinking of the copper‐modified poly(organosilazane) and subsequent pyrolysis lead to the copper‐containing ceramics. The copper is reduced to copper metal during the pyrolysis step up to 1000 °C, as observed by solid‐state ⁶⁵Cu NMR spectroscopy, SEM images, and energy‐dispersive spectroscopy (EDS). Powder diffraction experiments verified the presence of crystalline copper. All Cu@SiCN ceramics show catalytic activity towards the oxidation of cycloalkanes using air as oxidant. The selectivity of the reaction increases with increasing copper content. The catalysts are recyclable. This study proves the feasibility of this molecular approach to metal‐containing SiCN precursor ceramics by using silylaminopyridinato complexes. Furthermore, the catalytic results confirm the applicability of this new class of metal‐containing ceramics as catalysts.     

On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-christoffel rules, I

Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval. Gauss quadrature rules are designed for each of them. Various questions concerning their construction and application are studied, theoretically or experimentally. They are so efficient that they should be considered for the development of software for special functions. Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.     

Highly active titanium-based olefin polymerization catalysts bearing bulky aminopyridinato ligands

A series of titanium complexes have been prepared using either salt metathesis or amine elimination reactions. Reacting the potassium salt of Ap*H {Ap*H = N-(2,6-diisopropylphenyl)-[6-(2,4,6-triisopropylphenyl)pyridin-2-yl]amine} (1) with [TiCl(4)(THF)(2)] results in the formation of a nucleophilic ring-opening product of the coordinated tetrahydrofuran (THF) ligand [Ap*TiCl(2)(OC(4)H(8)Cl)] (7). Alkylation with benzylmagnesium chloride gave rise to the corresponding benzyl complex [Ap*TiBn(2)(OC(4)H(8)Cl)] (8). However, THF ring opening was overcome by adopting an amine elimination route instead of salt metathesis. Mono(aminopyridinato)titanium trichloro complexes were prepared in high yields using [(CH(3))(2)NTiCl(3)], together with the corresponding sterically demanding aminopyridine as the starting material. The synthesized complexes could then be alkylated selectively. These complexes were characterized by spectroscopic methods, and their behavior in olefin polymerization and copolymerization of ethene and propene was explored. These mono(aminopyridinato)titanium trichloro complexes are less active if activated with methylaluminoxane (MAO). However, the activity increases strongly if MAO is replaced by d-MAO ("dry methylaluminoxane"). The catalysts show moderate activity toward propene polymerization, while ethylene-propylene copolymers in high-productivity with separated propene units were observed. The catalysts are also highly active in the co- and terpolymerization of 2-ethylidenenorbornene (ENB) with ethylene or ethylene-propylene, together with a very good incorporation of ENB. In all cases, the activity increases with an increase in the steric bulk of the protecting ligand.     

Metal–Metal Distances at the Limit: Cr–Cr 1.73 Å – the Importance of the Ligand and its Fine Tuning

The synthesis and structure of a homobimetallic chromium complex is reported. The ligand used to stabilise the quintuply bonded metals is a sterically fine‐tuned guanidinate. A chromium–chromium bond length of 1.7293(12) Å was observed. It is the shortest metal–metal distance reported for a stable compound yet.     

Ammonolysis of 1,2‐Bis[dichloro(methyl)silyl]ethane. On the Structure of 1,3,6,8‐Tetramethyl‐2,7,11,12‐tetraaza‐1,3,6,8‐tetrasila‐tricyclo[6.2.1.13,6]‐dodecane

Ammonolysis of 1,2‐bis[dichloro(methyl)silyl]ethane afforded a crystalline tricyclic silazane along with polymeric material. The crystalline material could be isolated in pure state. It was analyzed by ¹H, ¹³C, ¹⁵N and ²⁹Si NMR spectroscopy in solution, by ¹³C, ¹⁵N and ²⁹Si MAS NMR spectroscopy in the solid state, as well as by single‐crystal and powder X‐ray diffraction. The title compound exists as a single isomer in solution, whereas in the solid state the presence of several modifications is indicated, in particular by the solid‐state MAS NMR spectra.     

Convergence acceleration from the point of view of linear programming

The best possible bounds for the sum of an alternating series with completely monotonic terms, when 2N terms have been computed, are determined. It is shown that their difference decreases exponentially withN. Various generalizations are indicated. The optimal application of Euler's transformation is also discussed. The error of that method also decreases exponentially, though the logarithmic decrement is only about 2/3 compared with the best possible error bounds.     

On Summation Formulas Due to Plana, Lindelöf and Abel, and Related Gauss-Christoffel Rules, III

This is the last part of a work that has been divided into three parts. It contains Sections 4 and 5. Part I and Part II were published in BIT, vol. 37. Some amendments to the earlier parts are given in an appendix of this part.The reader is referred to Part I for the abstract, the contents and the references of the whole work. A short list of references for this part is found at the end of this part. A few new references of interest for the earlier parts are included.