Oscillation theorems for differential equations with a nonlinear damping term

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity, and, as a consequence, our results apply to wider classes of nonlinear differential equations. Two illustrative examples are considered.     

A three‐dimensional framework of bis­[tris­(ethyl­enediamine)zinc] tetra­iodo­cadmate diiodide assisted by N—H⋯I hydrogen bonds

The title salt, [Zn(C₂N₂H₈)₃]₂[CdI₄]I₂, conventionally abbreviated [Zn(en)₃]₂[CdI₄]I₂, where en is ethyl­enediamine, contains discrete [Zn(en)₃]²⁺ cations and [CdI₄]²⁻ anions with distorted octa­hedral and nearly tetra­hedral geometries, respectively, as well as uncoordinated I⁻ ions. The cation and the free I⁻ anion lie on twofold rotation axes and the [CdI₄]²⁻ anion lies on a axis in the space group I2d. The structure exhibits numerous weak inter‐ionic hydrogen bonds of two types, viz. N—H⋯I⁻(free ion) and N—H⋯I([CdI₄]²⁻), which support the resulting three‐dimensional framework.     

Three polymorphs of 3‐(3‐phenyl‐1H‐1,2,4‐triazol‐5‐yl)‐2H‐1‐benzopyran‐2‐one formed from different solvents

The polymorphic study of 3‐(3‐phenyl‐1H‐1,2,4‐triazol‐5‐yl)‐2H‐1‐benzopyran‐2‐one, C₁₇H₁₁N₃O₂, was performed due to its potential biological activity and revealed three polymorphic modifications in the triclinic space group P, the monoclinic space group P2₁ and the orthorhombic space group Pbca. These polymorphs have a one‐column layered type of crystal organization. The strongest interactions between the molecules of the studied structures is stacking between π‐systems, while N—H…N and C—H…O hydrogen bonds link stacked columns forming layers as a secondary basic structural motif. C—H…π hydrogen bonds were observed between neighbouring layers and their role is the least significant in the formation of the crystal structure. Packing differences between the polymorphic modifications are minor and can be identified only using an analysis based on a comparison of the pairwise interaction energies.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

Structure and intermolecular interactions of the fully negative solvatochromic merocyanine in the crystal phase

Structural Chemistry - Herein we report an X-ray crystallographic characterization of tetramethine merocyanine based on benzimidazole and thiobarbituric acid, which shows negative solvatochromism...     

Synthesis of 5-Hydroxymethyl-8-methyl-3-(3-aryl-[1,2,4]oxadiazol-5-yl)-2H-pyrano[2,3-c]pyridin-2-ones and Their Esters

New 5-hydroxymethyl-8-methyl-3-(3-aryl-[1,2,4]oxadiazol-5-yl)-2H-pyrano-[2,3-c]pyridin-2-ones and their esters were synthesized. The structure of obtained compounds was determined through a complete ¹H NMR analysis.     

New Luminescent Lanthanide Tetrakis-Complexes NEt4[LnL4] Based on Dimethyl-N-Benzoylamidophosphate

New lanthanide dimethyl-N-benzoylamidophosphate (HL) based tetrakis-complexes NEt₄[LnL₄] (Ln³⁺=La, Nd, Sm, Eu, Gd, Tb, Dy) are reported. The complexes are characterized by means of NMR, IR, absorption, and luminescent spectroscopy as well as by elemental, X-Ray, and thermal gravimetric analyses. The phenyl groups of the four ligands of the complex anion are directed towards one side, while the methoxy groups are directed in the opposite side, which makes the complexes under consideration structurally similar to calixarenes. The effect of changing the alkali metal counterion to the organic cation NEt₄⁺ on the structure and properties of the tetrakis-complex [LnL4]^- is analyzed. The complexes exhibit bright characteristic for respective lanthanides luminescence. Rather high intensity of the band of ⁵D₀→⁷F₄ transition, observed in the luminescence spectrum of NEt₄[EuL₄], is discussed based on theoretical calculations.     

Conformational polymorphs of 3‐cyclopropyl‐5‐(3‐methyl‐[1,2,4]triazolo[4,3‐a]pyridin‐7‐yl)‐1,2,4‐oxadiazole

The dipharmacophore compound 3‐cyclopropyl‐5‐(3‐methyl‐[1,2,4]triazolo[4,3‐a]pyridin‐7‐yl)‐1,2,4‐oxadiazole, C₁₂H₁₁N₅O, was studied on the assumption of its potential biological activity. Two polymorphic forms differ in both their molecular and crystal structures. The monoclinic polymorphic form was crystallized from more volatile solvents and contains a conformer with a higher relative energy. The basic molecule forms an abundance of interactions with relatively close energies. The orthorhombic polymorph was crystallized very slowly from isoamyl alcohol and contains a conformer with a much lower energy. The basic molecule forms two strong interactions and a large number of weak interactions. Stacking interactions of the `head‐to‐head' type in the monoclinic structure and of the `head‐to‐tail' type in the orthorhombic structure proved to be the strongest and form stacked columns in the two polymorphs. The main structural motif of the monoclinic structure is a double column where two stacked columns interact through weak C—H…N hydrogen bonds and dispersive interactions. In the orthorhombic structure, a single stacked column is the main structural motif. Periodic calculations confirmed that the orthorhombic structure obtained by slow evaporation has a lower lattice energy (0.97 kcal mol^?1) compared to the monoclinic structure.