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

Darboux coordinates, Yang-Yang functional, and gauge theory

The moduli space of SL(2) flat connections on a punctured Riemann surface with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function of the variety of SL(2)-opers is identified with the universal part of the effective twisted superpotential of the corresponding four dimensional N=2 supersymmetric theory subject to the two-dimensional Omega-deformation. This allows to give a definition of the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group, and connect it to the instanton counting of the four dimensional gauge theories, in the rank one case.Comment: 25 pages, 11 figures, v1. in the proceedings of Cargese conference "String Theory: Formal Developments and Applications" (Jun 21-Jul 3, 2010); reported also at six other conferences in 2010, v2. references correcte

GEOMETRY OF TREFOIL CONE \u2013 MANIFOLD

In this paper we prove that Trefoil knot cone manifold T (\u3b1) with cone angle \u3b1 is spherical for \u3c0/3 < \u3b1 < 5\u3c0/3. We show also that its spherical volume is given by the formula Vol(T (\u3b1)) = (3\u3b1 12 \u3c0)2 /12

Chemical Oxidative Polymerization of Methylene Blue: Reaction Mechanism and Aspects of Chain Structure

The kinetic regularities of the initial stage of chemical oxidative polymerization of methylene blue under the action of ammonium peroxodisulfate in an aqueous medium have been established by the method of potentiometry. It was shown that the methylene blue polymerization mechanism includes the stages of chain initiation and growth. It was found that the rate of the initial stage of the reaction obeys the kinetic equation of the first order with the activation energy 49 kJ · mol−1. Based on the proposed mechanism of oxidative polymerization of methylene blue and the data of MALDI, EPR, and IR spectroscopy methods, the structure of the polymethylene blue chain is proposed. It has been shown that polymethylene blue has a metallic luster, and its electrical conductivity is probably the result of conjugation over extended chain sections and the formation of charge transfer complexes. It was found that polymethylene blue is resistant to heating up to a temperature of 440 K and then enters into exothermic transformations without significant weight loss. When the temperature rises above 480 K, polymethylene blue is subject to endothermic degradation and retains 75% of its mass up to 1000 K