Point approximation of a space-homogeneous transport equation

Summary We present an approximation method of a space-homogeneous transport equation which we prove is convergent. The method is very promising for numerical computation. Comparison of a numerical computation with an exact solution is given for the Master equation.     

Kinetics evidence for a complex between peroxynitrous acid and titanium(IV)

The reaction between TiO(2+) and ONOOH in 0.9 M H(2)SO(4) provides evidence for direct formation, previously unobserved, of a HOONO-metal complex. The reaction proceeds via formation of an end-on complex (k = 3.0 x 10(2) M(-1) s(-1)) that rearranges to form a side-on complex (k approximately equal to 20 s(-1)). With ONOOH in excess, this rearrangement proceeds more slowly (k approximately equal to 0.1 s(-1)), probably because multiple hydrogen oxoperoxonitrate molecules form end-on complexes with oxotitanium(IV) and hinder rearrangement to the side-on complex. The absorption spectrum of the final product is that of TiO(2)(2+). Presumably, during the rearrangement or later, NO+ is lost.     

Schmelzreaktionen mit Aluminiumchlorid. I. Mitteilung. Über Kondensationsreaktionen von 1-Aminoanthrachinon und 1 -Amino-anthrachinon-2-sulfonsäure in der Pyridin-Aluminiumchlorid-Schmelze

The condensation of 1-aminoanthraquinone to 1,1′-diamino-2,2′-dianthraquinonyl in a melt of anhydrous aluminium chloride and moist pyridine has been reinvestigated. By a similar procedure under anhydrous conditions the sodium salt of 1-aminoanthraquinone-2-sulfonic acid yielded the sodium salt of 4,4′-diamino-1,1′-dianthraquinonyl-3,3′-disulfonic acid.     

Solid-State Photocyclodimerization of 1-Thiocoumarin (= 2H-1-Benzothiopyran-2-one)

Irradiation (λ > 390 nm) of 2H-1-benzothiopyran-2-one ( 1 ) in the solid state affords selectively 6aα,6bα,12bα,12cα -tetrahydrocyclobuta[1,2-c:4,3-c′]bis[1]benzothiopyran -6,7-dione ( 2 ), the head-to-head (HH) cis-cisoid-cis-dimer, while irradiation of 1 in the solid state using shorter wavelengths (λ > 340 nm) affords a mixture of all four cis-fused tricyclic dimers 2 – 5 . These results represent a novel wavelength effect in solid-state photochemistry.     

Green chemistry and sustainability metrics in the pharmaceutical manufacturing sector

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Two weight commutators in the Dirichlet and Neumann Laplacian settings

In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian ${\mathrm{\Delta }}_{N_{+}}$ on the upper half space ${\mathbb{R}}_{+}^n$ and the reflection Neumann Laplacian ${\mathrm{\Delta }}_N$ on ${\mathbb{R}}^n$ with respect to the weights associated to ${\mathrm{\Delta }}_{N_{+}}$ and ${\mathrm{\Delta }}_N$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to ${\mathrm{\Delta }}_N$ is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt–Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via the area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on ${\mathbb{R}}^n$ and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt $A^p$ weights with $p\in (1,2]$ while the previously known related results cover only $p\in (1,\frac{n+1}{n}]$. We also point out that this two weight commutator theorem might not be true in the setting of general operators L, and in particular we show that it is not true when L is the Dirichlet Laplacian ${\mathrm{\Delta }}_{D_{+}}$ on ${\mathbb{R}}_{+}^n$.