Volume formula for a Z2\mathbb{Z}_2-symmetric spherical tetrahedron through its edge lengths

The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle $\pi$ in the middle points of a certain pair of its skew edges.Comment: 27 pages, 2 figures; enhanced and improved exposition, typos corrected; Arkiv foer Matematik, 201

Volumes of polytopes in spaces of constant curvature

We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a solution for Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find a few versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.Comment: 22 pages, 9 figures, 58 reference

Synthesis and spectral properties of fluorescent dyes based on 4-styryl-1,8-naphthalimide

The paper reports on synthesis and spectroscopic study of novel Nbutyll44styryll1,88naphthall imide dyes bearing methoxy (1), dimethoxy (2), and dimethylamino (3) groups in the styryl fragg ment. It is shown that all synthesized compounds demonstrate positive solvatochromism, high values of Stokes shift in polar solvents, and fluorescence in the long wavelength part of visible range. These facts indicate a potential application of these compounds as fluorescent dyes in the biochemistry. The changes in the dipole moments of the molecules caused by excitation were estimated using Lippert—Mataga equation. The obtained results could be assigned to the formation of the excited states with intramolecular charge transfer. The formation of the twisted states with charge transfer was suggested in the case of compound 3, while the fluorescence quantum yield was significantly reduced in polar protic solvents