Optimal error estimates of a locally one-dimensional method for the multidimensional heat equation

For the multidimensional heat equation in a parallelepiped, optimal error estimates inL ₂(Q) are derived. The error is of the order of +¦h¦² for any right-hand sidef L ₂(Q) and any initial function ; for appropriate classes of less regularf andu ₀, the error is of the order of ((+¦h¦² ), 1/2<1.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 185–197, August, 1996.     

Convergence of Krylov methods for sums of two operators

We discuss the convergence of Krylov subspace methods for equationsx =Tx +f whereT is a sum of two operators,T =B +K, whereB is bounded andK is nuclear. Bounds are given for inf Q _k (B+K) and for inf p _k (B+K), where the infimum is over all polynomials of degreek, such thatQ _k is monic andp _k is normalized:p _k (1) = 1.     

High-Resolution,Higher-Order UV/VIS Derivative Spectrophotometry

Derivative spectrophotometry has gained increasing importance in the past two years and is currently experiencing vigorous development. Following and introduction the present article provides a review of this extremely effective method. In particular, the advantages of higher-order derivative spectrophotometry (HODS method, n > 2) are discussed on the basis of practical examples from a wide range of analytical fields. The results are achieved with the aid of a newly developed analog computer unit, whereby for the first time readily reproducible, low-noise, on-line spectra can be obtained up to the 7th order and even, in favorable cases, up to the 9th order. In practice it has proved valuable to work with spectra of the 3rd to 5th order; but even higher derivations could be profitable for the separation of strongly superposed signals or for “fingerprinting”.     

Modern Analysis

The important advances being made in modern analytical methods are indicative of the fundamental changes that are occuring in the theory and practice of “analytical chemistry”. “Information optimization” demands a new approach in teaching and research, and calls for the intergration of chemistry with other scientific and technical disciplines.     

Catalyzed Addition of Aldehydes to Activated Double Bonds—A New Synthetic Approach

In the presence of cyanide ions as catalyst, aromatic and heterocyclic aldehydes can be smoothly added to α,β-unsaturated ketones, esters, and nitriles in aprotic solvents to form γ-diketones, 4-oxo carboxylic esters, and 4-oxo nitriles. Thiazolium salts in the presence of bases are also suitable catalysts; they permit not only addition of aromatic and heterocyclic aldehydes but also the addition of aliphatic aldehydes.     

A family of multiderivative methods for the numerical solution of the Schrödinger equation

A family of multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the Schrödinger equation. The methods are called multiderivative since uses derivatives of order two, four or six. Numerical application of the new obtained methods to the Schrödinger equation shows their efficiency compared with other similar well known methods of the literature.Active Member of the European Academy of Sciences and Arts     

The Synthesis of Unusual Organic Molecules Using Cyclic Peroxides. New synthetic methods (3)

The photodecarboxylation of malonyl peroxides into α-lactones^[1] and the thermal conversion of the 1,4-endo-peroxide 4,5-epoxy-3,6-epidioxy-1-cyclohexene into the novel benzene trioxide^[2] are two recent examples of the potential of cyclic peroxides in the synthesis of unusual organic molecules. The former transformation entails a fragmentation, the latter a rearrangement process. Most reported examples fall into one of these two gross reaction types. Of the numerous examples that have been reported in the literature during the last two decades, only those shall be focused on that lead to unusual compounds or constitute efficient syntheses of known compounds, in order to stress the convenience of cyclic peroxides in the synthesis of organic compounds.     

Intramolecular,Stoichiometric (Li,Mg, Zn) and Catalytic (Ni,Pd, Pt) Metallo-Ene Reactions in Organic Synthesis [New Synthetic Methods (75)]

Metallo-ene reactions, hardly recognized until very recently, have experienced a breathtaking development when applied in an intramolecular sense. Efficient regio- and stereoselective magnesium-ene cyclizations have served as a cornerstone for numerous syntheses of structurally diverse natural products (e.g., sesquiterpenes of marine or plant origin, alkaloids, fragrances, insect defense compounds, and a fungitoxin). A brilliant example is the synthesis of the elusive odorant (+)-khusimone which outshines 20 years of work in the field of tricyclovetivane synthesis. Palladium-, platinum-, and nickel-catalyzed versions of the metallo-ene reaction are in a comparatively early stage of exploration, but, nevertheless, reveal intriguing potential. Hence an almost 100% stereospecific C? O→C? ;Pd-→ C? C chirality transfer permits simple and selective, cis- or trans-annelation processes. The mild cyclization conditions are compatible with various functional groups, such as nitrogen moieties, which offer interesting perspectives for the preparation of heterocycles (e.g., alkaloids) difficult to obtain by other methods. Carbon monoxide insertion reactions of the cyclized σ-metal intermediates were shown to afford annelated cyclopentanones and cyclopentenones with concomitant stereocontrolled formation of four carbon–carbon bonds. These and other observations, highlighted in this article, provide a platform for further extensions and applications of this powerful method in organic synthesis.     

Integrated Dust Analysis by Physical Methods

Environmental analyses show that the air which we breathe, and which is so essential to life, is in general a mixture of gaseous, liquid, and solid components. The solid airborne particles, whose concentration, homogeneity, chemical composition, size, and shape can vary over wide ranges, and whose origin may be “natural” or “artificial” are referred to as “dust”. Dust particles can act, inter alia, as condensation nuclei, catalysts, and directly as hazardous materials. Unfortunately, we still know far too little about dust. Dust analysis is extremely difficult and challenging, even for modern analytical chemistry; it is still far from being fully automated. The simultaneous determination of as many “dust parameters” as possible, and particularly the synoptic consideration of all available data against a background of physicochemical and technological knowledge on the development, transformation, and effects of dust, are summarized as “integrated dust analysis”.     

Primary methods of measurement in chemical analysis

 Primary methods of measurement have a central function in metrology. They are an essential component in the realisation of the SI units and therefore are indispensable for establishing traceability of measurements of all kinds of physical quantities to the corresponding SI units. This is also true for chemical analysis. Gravimetry, titrimetry, coulometry, and isotope dilution mass spectrometry (IDMS) are evaluated with regard to their potential to be primary methods according to a general definition of primary methods recently given by the Comité Consultatif pour la Quantité de Matière (CCQM). Optical absorption spectrometry and methods based on colligative properties are also considered. A general scheme for establishing traceability of chemical measurements to the SI units using primary methods is discussed. Received: 17 April 1997 · Accepted: 9 August 1997