On the length of arithmetic progressions in linear combinations of S-units

Recent finiteness results concerning the lengths of arithmetic progressions in linear combinations of elements from finitely generated multiplicative groups have found applications to a variety of problems in number theory. In the present paper, we significantly refine the existing arguments and give an explicit upper bound on the length of such progressions.     

Staudinger and retro-Staudinger reactions. The dichloro-β-lactam moiety as a useful handle for the synthesis of 4-aryl-2H-1,3-benzothiazine 1,1-dioxides

The dichloro-β-lactam ring, obtained via Staudinger reaction of 4-aryl-2H-1,3-benzothiazines, proved to be a useful protecting strategy for the synthesis of 4-aryl-2H-1,3-benzothiazine 1,1-dioxides. After oxidation of the 1,1-dichloroazeto[2,1-c][1,3]-benzothiazin-2-ones, the thiazine ring could be recovered selectively and in good yield by treatment with base. Thus, novel 4-aryl-2H-1,3-benzothiazine 1,1-dioxides were obtained efficiently.     

Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Linear multistep methods (LMMs) applied to approximate the solution of initial value problems—typically arising from method-of-lines semidiscretizations of partial differential equations—are often required to have certain monotonicity or boundedness properties (e.g., strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). An LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work, we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams–Bashforth family (with 1 ≤ k ≤ 3 steps).     

Kolmogorov-Smirnov isometries of the space of generalized distribution functions

In this paper we describe the structure of surjective isometries of the space of all generalized probability distribution functions on ? with respect to the Kolmogorov-Smirnov metric.     

Synthesis of novel dihydro-2H-1,2,4-oxadiazin-5(6H)-ones

The title compounds 4 were synthesized via the acid-catalyzed reaction of α-(acylaminooxy)carboxylic acid amides 6 with carbonyl compounds, and controlled catalytic hydrogenation of the resulting 2-benzyloxy-carbonyldihydro-2H-1,2,4-oxadiazin-5(6H)-ones 7 (R⁶ = PhCH₂O).     

Synthesis of quinazolino-1,3-benzothiazine derivatives

The ring-closure reactions of N-(3,4-dimethoxyphenylthiomethyl)-2-nitrobenzamide derivatives 5a,b with phosphoryl chloride gave 4-(2-nitrophenyl)-2H1,3-benzothiazine derivatives 7a,b , which on reduction yielded 4-(2′-aminophenyl)-3,4-dihydro-2H-1,3-benzothiazines 8a,b. Reaction of these compounds with phosgene led to a new heterocyclic ring system, 6H,8H-quinazolino[3,4-c][1,3]benzothiazine derivatives 9a,b. The structures of the title compounds were proved via their ir and nmr (¹H, ¹³C) spectra.