1-Methyl-3-(4-chloro­benzo­yl)imidazo[1,2-a]pyridin-1-ium-2-olate

In the mol­ecule of the title compound, C15H11ClN2O2, the nine-membered heterobicycle is approximately planar [largest deviation from least-squares plane = 0.012 (2) Å] and forms a dihedral angle of 51.14 (8)° with the plane of the 4-chloro­phenyl group. There is a non-classical intra­molecular hydrogen bond between the pyridine α-H atom and the O atom of the benzoyl group. The crystal structure is stabilized by weak C—H⋯O and C—H⋯Cl inter­actions involving the ‘olate’ O atom and the Cl atom attached to the benzoyl group as acceptors

Variational principles of micromagnetics revisited

We revisit the basic variational formulation of the minimization problem associated with the micromagnetic energy, with an emphasis on the treatment of the stray field contribution to the energy, which is intrinsically non-local. Under minimal assumptions, we establish three distinct variational principles for the stray field energy: a minimax principle involving magnetic scalar potential and two minimization principles involving magnetic vector potential. We then apply our formulations to the dimension reduction problem for thin ferromagnetic shells of arbitrary shapes

4-(5-Amino-1H-1,2,4-triazol-3-yl)pyridinium chloride monohydrate

In the cation of the title compound, C7H8N5 +·Cl−·H2O, the mean planes of the pyridine and 1,2,4-triazole rings form a dihedral angle of 2.3 (1)°. The N atom of the amino group adopts a trigonal–pyramidal configuration. The N atom of the pyridine ring is protonated, forming a chloride salt. In the crystal, inter­molecular N—H⋯O, N—H⋯N, N—H⋯Cl and O—H⋯Cl hydrogen bonds link the cations, anions and water mol­ecules into layers parallel to the (1, 0, ) plane

2-[2-(1H-Imidazol-1-yl)-2-adamant­yl]phenol

In the title mol­ecule, C19H22N2O, the imidazole and benzene rings form a dihedral angle of 84.53 (5)°. In the crystal, classical inter­molecular O—H⋯N hydrogen bonds pair the mol­ecules into centrosymmetric dimers, and C—H⋯π inter­actions further link these dimers into columns propagated in [100]

Exact Self-consistent Particle-like Solutions to the Equations of Nonlinear Scalar Electrodynamics in General Relativity

Exact self-consistent particle-like solutions with spherical and/or cylindrical symmetry to the equations governing the interacting system of scalar, electromagnetic and gravitational fields have been obtained. As a particular case it is shown that the equations of motion admit a special kind of solutions with sharp boundary known as droplets. For these solutions, the physical fields vanish and the space-time is flat outside of the critical sphere or cylinder. Therefore, the mass and the electric charge of these configurations are zero.Comment: 17 pages, Submitted to the International Journal of Theoretical Physic

Tris(ethane-1,2-diamine)copper(II) bis­(trifluoro­acetate)

In the title complex, [Cu(H2NCH2CH2NH2)3](CF3COO)2, the environment of the Cu atom is distorted octa­hedral, formed by six N atoms from three chelating ethane-1,2-diamine ligands. The Cu—N distances range from 2.050 (2) to 2.300 (2) Å. This complex cation and the two trifluoro­acetate anions are connected by weak N—H⋯O and N—H⋯F hydrogen bonds, forming a three-dimensional framework. In both anions, the F atoms are disordered over two positions; in one the site-occupancy factors are 0.55 and 0.45, in the other the values are 0.69 and 0.31

4-Allyl-3-(2-methyl-4-quinol­yl)-1H-1,2,4-triazole-5(4H)-thione

In the title compound, C15H14N4S, the quinoline and triazole rings form a dihedral angle of 41.48 (7)°. In the crystal, adjacent mol­ecules are linked by N—H⋯N hydrogen bonds, forming chains along [100]

Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods