On the maximum size of a k-zero-free set in an abelian group

A subset A of elements in an abelian group G is called k-zero-free if the equation x ₁ + x ₂ + ... + x _k = 0 has no solution in A. A k-zero-free set A in G is called maximal if A ∪ {x} is k-zero-free for no x ∈ G\A. Some bounds for the maximum size of a k-zero free set are obtained. In particular, we determine the maximum speed of a k-zero-free arithmetic progression in the cyclic group Z _n and find the upper and lower bounds for the maximum size of a k-zero-free set in an abelian group G. We describe the structure of a maximal k-zero-free set A in the cyclic group Z _n provided that gcd(n, k) = 1 and k|A| ≥ n + 1.     

Model of micropolar thin shell oscillations

The paper gives the equations of a general applied two-dimensional theory for the dynamics of micropolar elastic thin shells with independent fields of motion and rotation that completely take into account all rotation-shift and related deformations. Problems on free and forced oscillations of micropolar elastic shells are studied on the basis of this general theory. Special features for the dynamic behavior of shells made of a micropolar elastic material are revealed on the basis of numerical analysis.     

On Laser-Induced High-Order Wave Mixing and Harmonic Generation in a Graphene Quantum Dot

JETP Letters - We present results of numerical investigations of high-order wave mixing/harmonic generation (HWM/HHG) with many-body interaction processes derived by a strong two-frequency circular...     

Concise Synthesis of (+)-[13C4]-Anatoxin-a by Dynamic Kinetic Resolution of a Cyclic Iminium Ion

An asymmetric total synthesis of [¹³C₄]-anatoxin-a ([¹³C₄]- 1 ) has been developed from commercially available ethyl [¹³C₄]-acetoacetate ([¹³C₄]- 15 ). The unique requirements associated with isotope incorporation inspired a new, robust, and highly scalable route, providing access to 0.110 g of this internal standard for use in the detection and precise quantification of anatoxin-a in freshwater. A highlight of the synthesis is a method that leverages a cyclic iminium ion racemization to achieve dynamic kinetic resolution in an enantioselective Morita–Baylis–Hillman (MBH) cyclization.     

Synthesis and Some Transformations of 5-(1,4-Benzodioxan-2-yl)-4-phenyl-4H-1,2,4-triazole-3-thiol

Russian Journal of Organic Chemistry - The title compound was synthesized by reaction of 1,4-benzodioxane-2-carbohydrazide with phenyl isothiocyanate, followed by cyclization of the...     

Hexafluoroisopropanol as a unique solvent for stereoselective iododesilylation of vinylsilanes

Stereoselective preparation of iodoalkenes from vinylsilanes is described. 1,1,1,3,3,3-hexafluoroisopropanol serves as a unique solvent that ensures high yields and stereoselectivities in the iododesilylation of a variety of functionalized substrates.     

Synthesis and structure of 1-(buta-1,3-dien-2-yl)pyrazoles and their behavior in Diels-Alder reactions

A convenient procedure has been developed for the synthesis of cis-and trans-isomeric 1-(buta-1,3-dien-1-yl)-1H-pyrazoles by reaction of the corresponding pyrazoles with β-methylacrolein diethyl acetal and subsequent 1,4-cleavage of the nucleophilic substitution products. The behavior of the title compounds in Diels-Alder reactions with maleic anhydride has been studied. According to the ¹H NMR data, 1-(buta-1,3-dien-1-yl)-1H-pyrazole is a mixture of cis and trans isomers. Butadienylpyrazoles having methyl groups in the pyrazole ring do not react with maleic anhydride.     

Enolate Stabilization by Anion–π Interactions: Deuterium Exchange in Malonate Dilactones on π‐Acidic Surfaces

Of central importance in chemistry and biology, enolate chemistry is an attractive topic to elaborate on possible contributions of anion–π interactions to catalysis. To demonstrate the existence of such contributions, experimental evidence for the stabilization of not only anions but also anionic intermediates and transition states on π‐acidic aromatic surfaces is decisive. To tackle this challenge for enolate chemistry with maximal precision and minimal uncertainty, malonate dilactones are covalently positioned on the π‐acidic surface of naphthalenediimides (NDIs). Their presence is directly visible in the upfield shifts of the α‐protons in the ¹H NMR spectra. The reactivity of these protons on π‐acidic surfaces is measured by hydrogen–deuterium (H–D) exchange for 11 different examples, excluding controls. The velocity of H–D exchange increases with π acidity (NDI core substituents: SO₂R>SOR>H>OR>OR/NR₂>SR>NR₂). The H–D exchange kinetics vary with the structure of the enolate (malonates>methylmalonates, dilactones>dithiolactones). Moreover, they depend on the distance to the π surface (bridge length: 11–13 atoms). Most importantly, H–D exchange depends strongly on the chirality of the π surface (chiral sulfoxides as core substituents; the crystal structure of the enantiopure (R,R,P)‐macrocycle is reported). For maximal π acidity, transition‐state stabilizations up to ?18.8 kJ mol^?1 are obtained for H–D exchange. The Brønsted acidity of the enols increases strongly with π acidity of the aromatic surface, the lowest measured pK_a=10.9 calculates to a ΔpK_a=?5.5. Corresponding to the deprotonation of arginine residues in neutral water, considered as “impossible” in biology, the found enolate–π interactions are very important. The strong dependence of enolate stabilization on the unprecedented seven‐component π‐acidity gradient over almost 1 eV demonstrates quantitatively that such important anion–π activities can be expected only from strong enough π acids.