Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation

The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlev\'{e} method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.

Tripodal Oxazolidine-N-Oxyl Diradical Complexes of Dy3+ and Eu3+

Two diradical complexes of the formula [LnRad2(CF3SO3)3] c (Ln(III) = Dy, Eu, Rad = 4,4-dimethyl-2,2-bis(pyridin-2-yl)-1,3-oxazolidine-3-oxyl) were obtained in air conditions. These are the first examples of diradical compounds of lanthanides and oxazolidine nitroxide. The complexes were characterized crystallographically and magnetically. Single crystal XRD analysis revealed that their coordination sphere is composed of three monodentate triflates and two tripodal Rad, which coordinate the central atom in a tridentate manner via two N atoms of the pyridine groups and the O atom of a nitroxide group. The LnO5N4 polyhedron represents a spherical capped square antiprism with point symmetry close to C4v. The data of static magnetic measurements are compatible with the presence of two paramagnetic ligands in the coordination sphere of the metal.

A Generalized Two-Dimensional Index to Measure the Degree of Deviation from Double Symmetry in Square Contingency Tables

The double symmetry model satisfies both the symmetry and point symmetry models simultaneously. To measure the degree of deviation from the double symmetry model, a two-dimensional index that can concurrently measure the degree of deviation from symmetry and point symmetry is considered. This two-dimensional index is constructed by combining two existing indexes. Although the existing indexes are constructed using power divergence, the existing two-dimensional index that can concurrently measure both symmetries is constructed using only Kullback-Leibler information, which is a special case of power divergence. Previous studies note the importance of using several indexes of divergence to compare the degrees of deviation from a model for several square contingency tables. This study, therefore, proposes a two-dimensional index based on power divergence in order to measure deviation from double symmetry for square contingency tables. Numerical examples show the utility of the proposed two-dimensional index using two datasets.

Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter

The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. For algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE and can be systematically computed. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq equation reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.

New insights into singularity analysis

In this work, we emphasize the use of singularity analysis in obtaining analytic solutions for equations for which standard Lie point symmetry analysis fails to make any lucid decision. We study the higher-dimensional Kadomtsev–Petviashvili, Boussinesq, and Kaup–Kupershmidt equations in a more general sense. With higher-order equations, there can be a commensurate number of resonances and when consistency for the full equation is examined at each resonance the constant of integration is supposed to vanish from the expression so that it remains arbitrary, but if there is an instance of this not happening, the consistency can be partially established by giving the offending constant the value from the defining equation. If consistency is otherwise not compromised, the equation can be said to be partially integrable, i.e., integrable on a surface of the complex space. Furthermore, we propose an approach that is meant to magnify the scope of singularity analysis for equations admitting higher values for resonances or positive leading-order exponent.

Hydrogen-bonding in mono-, di- and tetramethylammonium dihydrogenphosphites

The crystal structures of methylammonium and dimethylammonium dihydrogenphosphite (MA⋅H2PO3, I2/a and DMA⋅H2PO3, P 2 1 / c $P{2}_{1}/c$ ) are built of infinite chains of hydrogen bonded H 2 P O 3 − ${\mathrm{H}}_{\mathrm{2}}\mathrm{P}{\mathrm{O}}_{\mathrm{3}}^{-}$ anions. The chains are connected by the ammonium cations via hydrogen bonding to di- (DMA⋅H2PO3) and triperiodic (MA⋅H2PO3) networks. Tetramethylammonium dihydrogenphosphite monohydrate (TMA⋅H2PO3⋅H2O) features temperature dependent dimorphism. The crystal structure of the high-temperature (HT, cubic P213) and low-temperature (LT, orthorhombic P212121) phases were determined at 150 and 100 K, respectively. The hydrogen bonding network in the HT phase is disordered, with H 2 P O 3 − ${\mathrm{H}}_{\mathrm{2}}\mathrm{P}{\mathrm{O}}_{\mathrm{3}}^{-}$ and H2O being located on a threefold axis and is ordered in the LT phase. On cooling, the point symmetry is reduced by an index of 3. The lost symmetry is retained as twin operations, leading to threefold twinning by pseudo-merohedry. The hydrogen-bonding networks of the HT and LT phases can be represented by undirected and directed quotient graphs, respectively.