Subharmonic solutions for nonautonomous sublinear second order Hamiltonian systems

AbstractSome existence theorems are obtained for subharmonic solutions of nonautonomous second order Hamiltonian systems by the minimax methods in critical point theory

Periodic Solutions of a Class of Non-autonomous Second-Order Systems

AbstractSome existence theorems are obtained by the least action principle for periodic solutions of nonautonomous second-order systems with a potential which is the sum of a subconvex function and a subquadratic function

2,2′-Dimethyl-1,1′-[2,2-bis­(bromo­methyl)propane-1,3-di­yl]dibenzimidazole hemihydrate

The title compound, C21H22Br2N4·0.5H2O, contains two benzimidazole groups which may provide two potential coordination nodes for the construction of metal–organic frameworks. The mean planes of the two imidazole groups are almost perpendicular, with a dihedral angle of 83.05 (2)°, and adjacent mol­ecules are linked into a one-dimensional chain by π–π stacking inter­actions between imidazole groups of different mol­ecules [centroid-to-centroid distances of 3.834 (2) and 3.522 (2) Å]

3-Carb­oxy­quinolin-1-ium-2-carboxyl­ate monohydrate

The title compound, C11H7NO4·H2O, contains a 3-carb­oxy­quinolin-1-ium-2-carboxyl­ate (qda) zwitterion and one water mol­ecule. In the crystal, pairs of N—H⋯O hydrogen bonds link the mol­ecules into inversion dimers, and these dimers are further connected by O—H⋯O hydrogen bonds into a three-dimensional supra­molecular architecture. In addition, π–π inter­actions occur between pyridine and benzene rings from different qda ligands [centroid–centroid distance = 3.749 (1) Å] and the dihedral angles of the –CO2H and –CO2 groups to the quinoline system are 8.47 (3) and 88.16 (6)°, respectively

Poly[(μ2-quinoline-3-carboxyl­ato-κ2 N:O)(μ2-quinoline-3-carboxyl­ato-κ3 N:O,O′)cadmium]

In the title compound, [Cd(C10H6NO2)2]n, the CdII atom is coordinated by three O atoms and two N atoms from four quinoline-3-carboxyl­ate (L −) ligands, leading to a distorted trigonal–bipyramidal geometry. The L − ligands link the CdII atoms into a plane parallel to (100), with one ligand being tridentate, coordinating via the N atom and chelating a second Cd atom, and the other being bidentate, bridging two Cd atoms via the N and one O atom.. This two-dimensional network extends into a double-layer network by π–π inter­actions, with centroid–centroid distances of 3.680 (2) and 3.752 (2) Å. Another type of π–π inter­action between pyridine rings [centroid–centroid distance = 3.527 (2) Å] leads to a three-dimensional supra­molecular architecture

Homoclinic orbits for a class of second-order Hamiltonian systems with concave–convex nonlinearities

In this paper, we study the existence of multiple homoclinic solutions for the following second order Hamiltonian systems \begin{equation*} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \end{equation*} where $L(t)$ satisfies a boundedness assumption which is different from the coercive condition and $W$ is a combination of subquadratic and superquadratic terms

Synergy of Pd atoms and oxygen vacancies on In₂O₃ for methane conversion under visible light

Methane (CH4) oxidation to high value chemicals under mild conditions through photocatalysis is a sustainable and appealing pathway, nevertheless confronting the critical issues regarding both conversion and selectivity. Herein, under visible irradiation (420 nm), the synergy of palladium (Pd) atom cocatalyst and oxygen vacancies (OVs) on In2O3 nanorods enables superior photocatalytic CH4 activation by O2. The optimized catalyst reaches ca. 100 μmol h-1 of C1 oxygenates, with a selectivity of primary products (CH3OH and CH3OOH) up to 82.5%. Mechanism investigation elucidates that such superior photocatalysis is induced by the dedicated function of Pd single atoms and oxygen vacancies on boosting hole and electron transfer, respectively. O2 is proven to be the only oxygen source for CH3OH production, while H2O acts as the promoter for efficient CH4 activation through ·OH production and facilitates product desorption as indicated by DFT modeling. This work thus provides new understandings on simultaneous regulation of both activity and selectivity by the synergy of single atom cocatalysts and oxygen vacancies

Bis(1,10-phenanthroline-5,6-dione-κ2 N,N′)silver(I) 2-hy­droxy-3,5-dinitro­benzoate

In the cation of the title salt, [Ag(C12H6N2O2)2](C7H3N2O7), the AgI atom is coordinated in a distorted tetra­hedral geometry by four N atoms from two 1,10-phenanthroline-5,6-dione ligands, while the 3,5-dinitro­salicylate anion has only a short contact [2.847 (6) Å] between one of its O atoms and the AgI atom. The dihedral angle between the two 1,10-phenanthroline-5,6-dione ligands is 58.4 (1)°. There is an intra­molecular O—H⋯O hydrogen bond in the 3,5-dinitro­salicylate anion

cyclo-Tetra­kis{μ-2,2′-dimethyl-1,1′-[2,2-bis­(bromo­meth­yl)propane-1,3-di­yl]di(1H-benzimidazole)-κ2 N 3:N 3′}tetra­kis­[bromidocopper(I)]

The title compound, [Cu4Br4(C21H22Br2N4)4], features a macrocyclic Cu4 L 4 ring system in which each CuI atom is coordinated by one bromide ion and two N atoms from two 2,2′-dimethyl-1,1′-[2,2-bis­(bromo­meth­yl)propane-1,3-di­yl]di(1H-benzimidazole) (L) ligands in a distorted trigonal–planar geometry. The L ligands adopt either a cis or trans configuration. The asymmetric unit contains one half-mol­ecule with the center of the macrocycle located on a crystallographic center of inversion. Each bromide ion binds to a CuI atom in a terminal mode and is oriented outside the ring. The macrocycles are inter­connected into a two-dimensional network by π–π inter­actions between benzimid­azole groups from different rings [centroid–centroid distance = 3.803 (5) Å