Let V be a nonsingular projective surface of Kodaira dimension κ(V)≥0. Let D be a reduced, effective, nonzero divisor on V with only simple normal crossings. In the present article, a pair (V, D) is said to be a minimal logarithmic surface of general type, if, by definition, K_V+D is a numerically effective divisor of self intersection number (K_V+D)²>0 and if K_V+D has positive intersection with every exceptional curve of the first kind on V. Here K_V is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka—Yau type inequality (\bar{c}₁²):=(K_V+D)²≤ 3\bar{c}₂:=3c₂(V)−3e(D). On the other hand, we can easily obtain (\bar{c}₁²)≥\frac{1}{15}\bar{c}₂−\frac{8}{5} by making use of [8; Theorem 5.5]. In the present article, we shall prove that (\bar{c}₁²)≥\frac{1}{9}\bar{c}₂−2 provided that the rational map Φ_|KV+D| defined by the complete linear system |K_V+D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus \bar{p}_g:=h⁰(V, K_V+D)=½(\bar{c}₁²)+2=3, D is an elliptic curve and V is the canonical resolution in the sense of Horikawa associated with a double covering h: Y→P². In addition, the branch locus B of h is a reduced curve of degree eight and the singular locus Sing B consists of points of multiplicity ≤ 3 except for at most one “simple quadruple point”.

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We define *-Hopf algebras Fun(SL_q(N, \mathbb{C};ε₁, ..., ε_N)), Fun(O_q(N, \mathbb{C};ε₁, ..., ε_N)) and Fun(Sp_q(n, \mathbb{C};ε₁, ..., ε_2n)) as the real complexifications of *-Hopf algebras Fun(SU_q(N;ε₁, ..., ε_N)), Fun(O_q(N;ε₁, ..., ε_N)) and Fun(Sp_q(n;ε₁, ..., ε_2n)) of [RTF] (for q>0). Such construction can be done for each coquasitriangular (CQT) *-Hopf algebra A. The obtained object A^\mathbb{CR} is also a CQT *-Hopf algebra. We describe the theory of corepresentations of A^\mathbb{CR} in terms of such a theory for A.

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We shall provide a large class of extremal positive maps in M₃(C) which are neither 2-positive nor 2-copositive and study the algebraic structure of the set of all positive linear maps in M₃(C).

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Let be given a dense domain \mathscr{D} in a Hilbert space and a closed symmetric operator T with domain containing \mathscr{D}. Then the restriction of T to \mathscr{D} generates (algebraically) two partial *-algebras of closable operators (called weak and strong), possibly nonabelian and nonassociative. We characterize them completely. In particular, we examine under what conditions they are regular, that is, consist of polynomials only, and standard. Simple differential operators provide concrete examples of all the pathologies allowed by the abstract theory.

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The tensor product of two representations of the discrete series and the limit of the discrete series of \mathscr{U}_q(su(1, 1)) is decomposed into the direct sum of irreducible components of \mathscr{U}_q(sl(1, 1)), and the Clebsch-Gordan coefficients with respect to this decomposition are computed in two ways. In some cases, the tensor product of an irreducible unitary representation of \mathscr{U}_q(su(2)) and a representation of the discrete series of \mathscr{U}_q(su(1, 1)) is decomposed into the direct sum of irreducible components of \mathscr{U}_q(sl(2)), and the Clebsch-Gordan coefficients with respect to this decomposition are calculated, too. Making use of these coefficients, the linearization formula of the matrix elements is obtained.

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A new deformation of SL(2, C) (considered as a real Lie group) is constructed and shown to have a Gauss type decomposition. The groups entering this decomposition are identified as E_μ(2) and its Pontryagin dual \hat{E}_μ(2). The whole group is the double group built over E_μ(2).

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In the present paper, existence, uniqueness and convergence properties of a discrete cubic spline which satisfies certain averaging interpolatory condition are established. This type of interpolatory condition has been studies earlier for usual cubic splines in [3]. More precise range for weights involved in the interpolatory condition and sharper error estimates than in [3], are obtained in the present paper.

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We show a Goldman's type theorem for any inclusions of (not necessarily AFD) factors with the principal graph A₅. Our main tools are correspondences and sectors.

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