Regularisierte Faltung von Distributionen. Teil 1: Zur Berechnung von Fundamentallösungen

Zusammenfassung Für 2 Distributionen, deren Faltung nicht existiert, wird—mehrdeutig—eine bezüglich eines Differentialoperators regularisierte Faltung definiert.An 4 Beispielen wird die Anwendbarkeit der regularisierten Faltung bei der Berechnung von Fundamentallösungen faktorisierbarer Differentialoperatoren gezeigt. Dabei wurde die Fundamentallösung in Beispiel 3.2. erstmalig ohne Überlegungen physikalischer Art hergeleitet. Beispiel 3.3. verallgemeinert bekannte Ergebnisse, wobei diese als Spezialfälle erscheinen (Bemerkung 3) oder als fehlerhaft nachgewiesen werden (Bemerkung 4). Die Fundamentallösungen der Beispiele 3.1 und 3.4 scheinen in dieser Form neu zu sein.Schließlich folgen einige einfache Sätze, mittels derer Fundamentallösungen von iterierten Differentialoperatoren angegeben werden können. Sie werden im 2. Teil der Arbeit (Eine Tabelle von Fundamentallösungen) angewendet. Literaturhinweise finden sich am Ende von Teil 2.

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Summary For two distributions (whose convolution does not necessarily exist) we define a (manyvalued) convolution regularized with respect to a differential operator.We illustrate by four examples the usefulness of the concept of a regularized convolution for computing fundamental solutions of factorisable differential operators. The fundamental solution of example 3.2 is derived for the first time without considerations from physics. Example 3.3 generalizes well-known results, which either appear as special cases (remark 3) or are proved to contain errors (remark 4). The fundamental solutions of examples 3.1 and 3.4 seem to be new in this form.Finally we give some simple propositions concerning fundamental solutions of iterated differential operators. They will be used in part 2 of this paper (a table of fundamental solutions)The references are given at the end of part 2.

     

Optimization of an electrolytic conductivity detector for determination of toxic nitrogen-containing food contaminants separated by open tubular column gas chromatography

The combination of open tubular column gas chromatography with electrolytic conductivity detection has been evaluated for the determination of nitrogen-containing pesticide residues in food extracts. Optimization of the column position at the column-detector interface was crucial to the successful operation of the detector. The signal-to-noise ratio and response stability of the detector are greatly influenced by the composition of the electrolyte solvent. Large volume splitless injections using retention gaps and optimized detector operating conditions enabled pesticide residues in food extracts to be determined at sub parts-per-million levels. Although the electrolytic conductivity detector is less sensitive than the thermionic ionization detector, its greater nitrogen selectivity can he crucial to the determination of nitrogen-containing contaminants in food extracts, particularly in complex mixtures where phosphorus-containing contaminants or matrix compounds are also present.     

Hydrogen‐bonding patterns in rac‐1‐acetyl‐5‐methyl‐2‐thioxoimidazolidin‐4‐one

In the title compound, C₆H₈N₂O₂S, also known as N‐acetyl‐2‐thiohydantoin–alanine, the molecules are joined by N—H...O hydrogen bonds, forming centrosymmetric R₂²(8) dimers; these dimers are linked by C—H...O interactions to form R₂²(10) rings, thus forming C₂²(10) chains that run along the [101] direction.     

The reaction of aromatic α,β-unsaturated ketones with 4,5-diamino-1,6-dihydropyrimidin-6-ones

The reaction of 4,5-diamino-1,6-dihydropyrimidin-6-ones 1 with one equivalent of the chalcones 2 leads in an acidic medium to the formation of the 2,4-diaryl-2,3,6,7-tetrahydro-1H-pyrimido[4,5-b][1,4]diazepin-6-ones 3a-m . The structure elucidation of the products is based on detailed nmr investigations including selective ¹³C[¹H] decoupling experiments.     

A New Strategy for the Synthesis of alpha-Difluoromethyl-Substituted alpha-Hydroxy and alpha-Amino Acids

A new method for the preparation of alpha-chlorodifluoromethyl-, alpha-bromodifluoromethyl-, and alpha-difluoromethyl-substituted alpha-hydroxy and alpha-amino acid esters 11, 19-21 is described. The key step of the synthesis is the regioselective alkylation of ketones 5, 7-9 and imines 16-18 with C-nucleophiles. The ketones 7-9 are readily available from 3,3,3-trifluorolactate 1 by a five-step procedure. Subsequent removal of the protecting groups from 19-21 provides the corresponding free amino acids 25, 26, 28.     

Photochemical reactions. 146th communication. Photochemistry of conjugated methano-epoxydienes: Participation of the neighboring cyclopropane ring in product formation via carbonyl ylide and carbene intermediates

On singlet excitation (λ = 254 nm, THF, pentane or hexane), the diastereoisomeric methano-epoxydienes (E)- 6 and (E)- 7 undergo interconversion and yield the products 8 – 11 . The main process is the cleavage of the oxirane ring to the vinyl carbene intermediate e which undergoes addition to the adjacent double bond furnishing the cyclopropene 8 . The alternative carbene intermediate f is evidenced by the formation of the cyclobutene 10 . For the fragmentation leading to 11 , the carbene f as well as the dipolar species h are considered as possible intermediates. On triplet sensitization (acetone, λ > 280 nm), (E)- 7 shows behavior typical of epoxydienes, undergoing fission of the C? O bond of the oxirane ring and isomerization to the compounds 13 , 14 and (E/Z)- 15 .     

The invariance principle for ϕ-mixing sequences

Summary In this paper we investigate the invariance principle for -mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For -mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.     

Pyridazines. LIX. Synthesis of c-annelated pyridazines from 3-amino-4-pyridazinecarbonitrile

The utility of the aminonitrile 1 as an educt for the preparation of several new examples of heterocyclefused pyridazines (the [1,2,4]triazolo[1′,5′:1,6]pyrimido[4,5-c]pyridazine 7 , the pyrimido[4,5-c]pyridazines 8, 10a,b , and the pyrido[2,3-c]pyridazine 11 ) is demonstrated.     

Identification of cis/trans isomers of methyl ester and oxazoline derivatives of unsaturated fatty acids using GC–FTIR–MS

Identification of cis/trans isomers of unsaturated fatty acids cannot usually be achieved by GC-MS (gas chromatography-mass spectrometry) without reference substances. In this study a GC-FTIR-MS system (gas chromatography-Fourier transform-mass spectrometry) was used to identify fatty acid methyl esters (FAMEs) and differentiate between the cis/trans isomers. Besides methyl esters, 2-alkenyl-4,4-dimethyloxazoline derivatives (DMOX), which have been used to locate double bond positions of unsaturated fatty acids, were examined with respect to their suitability for cis/trans differentiation. A combined GC-FTIR-MS system with a wide band (4000–550 cm^?1) mercury cadmium telluride (MCT) detector was used in series and parallel to identify 31 reference unsaturated fatty acids, including 7 pairs of cis/trans isomers. Serum samples of healthy persons and commercially available fish oil were analyzed as examples of complex mixtures. Using splitless injection the detection limit for the less sensitive IR detector was 25 ng/μl in case of the weak cis and trans bands. In the FTIR spectra cis/trans isomers were identified by analysis of bands arising from C? H out-of-plane (oop) bending: for both the FAME and DMOX derivatives cis-1,2-disubstituted double bonds give a strong band near 720 cm^?1 and the corresponding trans isomers near 967 cm^?1. cis Isomers could be identified further by a band at 3012 cm^?1. With the combined data of the GC-FTIR-MS system it is now possible to identify polyunsaturated fatty acids with regard to the discrimination of cis/trans isomers.