Universal algebras

A survey of the development of the theory of universal algebras in the years 1976–1988. Such basic directions of development of this theory as classes of universal algebras (varieties, quasivarieties, etc.), Mal'tsev conditions, congruences, sheaves, homomorphisms, subalgebras, products of algebras, clones of operations, completeness, systems of equations in algeberas, and connections with computer mathematics show up in it. Considerable attention is given to concrete classes of algebras (p-algebras, BCK-algebras, unary algebras, etc.).Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 27, pp. 45–124, 1989.     

Local homology and dimensional full-valuedness

A natural criterion for dimensional full-valuedness of locally compact spaces with finitely generated local homology is given.     

Paraelectric resonance of centers with icosahedral symmetry

The paraelectric resonance spectrum of non-central impurity ions situated in crystal fields of icosahedral symmetry is predicted. A multiwell potential is considered that corresponds to the displacement of the ions from the center of symmetry toward the vertices of the icosahedron, the faces of a dodecahedron, and the pentagonal faces of a fullerene molecule. The effective Hamiltonian of endohedral complexes in external electric fields with arbitrary orientation is obtained, its eigenvalues are found, and the line intensities of different transitions between the tunnelling levels are calculated. Fiz. Tverd. Tela (St. Petersburg) 39, 1693–1698 (September 1997)     

Modifications based on coumarins. II. Synthesis ofα-monoglycerides of karatavic and galbanic acids and their phosphates

The synthesis has been performed of -monoglyceride derivatives of coumarins — karatavic and galbanic acids and their modified derivatives — phosphates. The structures of the compounds have been confirmed by IR, UV, PMR, and¹³C and³¹P NMR spectroscopies.A. B. Bekturov Institute of Chemical Sciences, Academy of Sciences of the Republic of Kazakhstan, Almaty Translated from Khimiya Prirodnykh Soedinenii, No. 2, pp. 217–220, March–April, 1997.     

Properties of semisimple Hopf algebras

Semisimple finite-dimensional Hopf algebras all of whose non-one-dimensional irreducible modules of the same dimension are isomorphic are considered. The one-dimensional submodules of the tensor squares of irreduciblemodules are indicated and the almost cocommutative Hopf algebras and the left ideal coideals are described.     

The number of additional singular points in the Riemann-Hilbert problem on a Riemann surface

We present an upper bound for the number of additional singular points that are sufficient to construct a system of linear equations with given regular singular points and a given monodromy on a Riemann surface.