Conservation Laws of The Generalized Riemann Equations

Two special classes of conserved densities involving arbitrary smooth functions are explicitly formulated for the generalized Riemann equation at arbitrary N. The particular case when N = 2 covers most of the known conserved densities of the equation, and the result is also extended to the famous Gurevich-Zybin, Monge-Ampere and two-component Hunter-Saxton equations by considering certain reductions.     

Coordinate transformation and construction of finite element mesh in a diverted tokamak geometry

A coordinate transformation technique between straight magnetic field line coordinate system (Ψ, θ) and Cartesian coordinate system (R, Z) is presented employing a Solov'ev solution of the Grad-Shafranov equation. Employing the equilibrium solution, the poloidal magnetic flux Ψ(R, Z) of a diverted tokamak, magnetic field line equation is solved computationally to find curves of constant poloidal angle θ, which provides us with explicit relations R = R(Ψ, θ) and Z = Z(Ψ, θ). Correspondingly, conversion from one coordinate to the other along particle trajectories in the vicinity of separatrix is demonstrated. Based on the magnetic structure, a finite element mesh is generated in a diverted tokamak geometry to solve Poisson's equation.     

Synthesis and Diverse Transformations of a Dinitrogen Dititanium Hydride Complex Bearing Rigid Acridane-Based PNP-Pincer Ligands

Studies on N₂ activation and transformation by transition metal hydride complexes are of particular interest and importance. The synthesis and diverse transformations of a dinitrogen dititanium hydride complex bearing the rigid acridane-based ^acriPNP-pincer ligands {[(^acriPNP)Ti]₂(μ₂-η¹:η²-N₂)(μ₂-H)₂} are presented. This complex enabled N₂ cleavage and hydrogenation even without additional H₂ or other reducing agents. Furthermore, diverse transformations of the N₂ unit with a variety of organometallic compounds such as ZnMe₂, MgMe₂, AlMe₃, B(C₆F₅)₃, PinBH, and PhSiH₃ have been well established at the rigid ^acriPNP-ligated dititanium framework, such as reversible bonding-mode change between the end-on and side-on/end-on fashions, diborylative N=N bond cleavage, the formal insertion of two dimethylaluminum species into the N=N bond, and the formal insertion of two silylene units into the N=N bond. This work has revealed many unprecedented aspects of dinitrogen reaction chemistry.     

Reversible Transformation between Cubane and Stairstep Cu4I4 Clusters Using Heat or Solvent Vapor

The controlled self‐assembly of CuI and an asymmetric ligand with mixed N/S donors, 2‐(tert‐butylthio)‐N‐(pyridin‐3‐yl)acetamide ( L ), afforded three Cu^I coordination polymers (CPs), [Cu₄I₄ L ₂(MeCN)₂]_n ( 1 ), [Cu₄I₄ L ₂]_n ( 2 ), and {[Cu₄I₄ L ₂] ? MeOH}_n ( 3 ). X‐ray analyses showed that CPs 1 – 3 are supramolecular isomers with 1, 2, and 3D structures, respectively. CP 1 adopts a stairstep Cu₄I₄ cluster, whereas CPs 2 and 3 are composed of cubane‐like Cu₄I₄ clusters. Crystal‐to‐crystal transformations of 1 to 2 and 3 showed reversible transformations between different Cu₄I₄ clusters using heat or solvent (acetonitrile or methanol) vapor. CP 2 was reversibly transformed to 3 by the addition of methanol and heat. Therefore, the transformations between supramolecular isomers 1 , 2 , and 3 are completely reversible.     

Synthesis of N-aryl-hexaoxazadispiroalkanes using lanthanide catalysts

An efficient method was developed for the synthesis of N-arylhexaoxazadispiroalkanes by the ring transformation of heptaoxadispiroalkanes with arylamines in the presence of lanthanide salts as catalysts.     

Polymorphic phase transformations of 3‐chloro‐trans‐cinnamic acid and its solid solution with 3‐bromo‐trans‐cinnamic acid

We have investigated the polymorphic phase transformations above ambient temperature for 3‐chloro‐trans‐cinnamic acid (3‐ClCA, C₉H₇ClO₂) and a solid solution of 3‐ClCA and 3‐bromo‐trans‐cinnamic acid (3‐BrCA, C₉H₇BrO₂). At 413 K, the γ polymorph of 3‐ClCA transforms to the β polymorph. Interestingly, the structure of the β polymorph of 3‐ClCA obtained in this transformation is different from the structure of the β polymorph of 3‐BrCA obtained in the corresponding polymorphic transformation from the γ polymorph of 3‐BrCA, even though the γ polymorphs of 3‐ClCA and 3‐BrCA are isostructural. We also report a high‐temperature phase transformation from a γ‐type structure to a β‐type structure for a solid solution of 3‐ClCA and 3‐BrCA (with a molar ratio close to 1:1). The γ polymorph of the solid solution is isostructural with the γ polymorphs of pure 3‐ClCA and pure 3‐BrCA, while the β‐type structure produced in the phase transformation is structurally similar to the β polymorph of pure 3‐BrCA.     

Formation of Acetylene in the Reaction of Methane with Iron Carbide Cluster Anions FeC3− under High‐Temperature Conditions

The underlying mechanism for non‐oxidative methane aromatization remains controversial owing to the lack of experimental evidence for the formation of the first C?C bond. For the first time, the elementary reaction of methane with atomic clusters (FeC₃^?) under high‐temperature conditions to produce C?C coupling products has been characterized by mass spectrometry. With the elevation of temperature from 300 K to 610 K, the production of acetylene, the important intermediate proposed in a monofunctional mechanism of methane aromatization, was significantly enhanced, which can be well‐rationalized by quantum chemistry calculations. This study narrows the gap between gas‐phase and condensed‐phase studies on methane conversion and suggests that the monofunctional mechanism probably operates in non‐oxidative methane aromatization.     

Tractable Partially Ordered Sets Derived from Root Systems and Biased Graphs

We study new posets Q obtained by removing from a geometric lattice L ofa biased graph certain flats indexed by a simplicial complex . (One example of L is the lattice of flats of thevector matroid of a root system B _n .) We study the structureand compute the characteristic polynomial of Q. With certainchoices of L and , including ones for which Q is alattice interpolating between those of B _n and D _n , we observe curious relationships among the roots of thecharacteristic polynomials of Q, L, and .     

Hierarchical ferromagnetic vector spin model possessing the Lee-Yang property. Thermodynamic limit at the critical point and above

The hierarchical ferromagneticN-dimensional vector spin model as a sequence of probability measures onR ^N is considered. The starting element of this sequence is chosen to belong to the Lee-Yang class of measures that is defined in the paper and includes most known examples (⁴ measures, Gaussian measures, and so on). For this model, we probe two thermodynamic limit theorems. One of them is just the classical central limit theorem for weakly dependent random vectors. It describes the convergence of classically normed sums of spins when temperature is sufficiently high. The other theorem describes the convergence of more than normally normed sums that holds for some fixed temperature. It corresponds to the strong dependence of spins, which appears at the critical point of the model.