Flat affine spheres in R 3

Nondegenerate affine surfaces in R ³ which are affine spheres and have flat affine metrics are classified. Those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defined by (x ²+y ²)z=1.Partially supported by NSF Grant DMS 8802664.Partially supported by NSERC Operating Grant A2501.     

On duality gap in binary quadratic programming

We investigate in this paper the duality gap between the binary quadratic optimization problem and its semidefinite programming relaxation. We show that the duality gap can be underestimated by ${\xi_{r+1}\delta^2}$ , where ?? is the distance between {?1, 1} ⁿ and certain affine subspace, and ?? _r+1 is the smallest positive eigenvalue of a perturbed matrix. We also establish the connection between the computation of ?? and the cell enumeration of hyperplane arrangement in discrete geometry.     

On right congruences of semigroups having no proper essential right congruences

A right congruence ?? in a semigroup S is essential if for any right congruence ?? we have ??????=?? (the identity relation) implies ??=??. Clearly, the universal relation, ??, is an essential right congruence. We say ?? is proper if ??????. In this paper we get a necessary and sufficient condition for a semigroup with an identity element?1 and having no proper essential right congruences to have a distributive lattice of right congruences.     

On locally strongly convex affine hypersurfaces with parallel cubic form. Part I

In this paper, we study locally strongly convex affine hypersurfaces of Rn+1 that have parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric; it is known that they are affine spheres. In dimension n?7 we give a complete classification of such hypersurfaces; in particular, we present new examples of affine spheres.     

Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces

In this paper, we introduce the weakly relaxed ??-pseudomonotone mapping which generalizes the class of pseudomonotone mappings. Some examples are given which show that the weakly relaxed ??-pseudomonotonicity is a proper generalization of pseudomonotonicity for both scalar and vector valued bi-functions. Using the KKM technique, we establish the existence of solutions of equilibrium problems and vector equilibrium problems with weakly relaxed ??-pseudomonotonicity in the reflexive Banach spaces. The present work extends some corresponding results of the variational-like inequalities (Fang and Huang in J. Optim. Theory Appl. 118:327?C338, 2003; Lee and Lee in J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 11: 231?C242, 2004) to equilibrium problems.     

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps.     

Semigroups related to certain group presentations

For every prime p, we study semigroups defined by the presentations $$\pi_{1}= \bigl\langle a,b,c \mid ab=bac,\ bc=cb,\ ac=ca,\ a^{p^{\alpha}+1}=a, b^{p^{\beta}+1}=b,\ c^{p^{\gamma}+1}=c \bigr\rangle, $$ where ????????????1 and $$\pi_{2}= \bigl\langle a,b\bigm{|} ab=ba^{1+p^{\alpha-\gamma}},\ a^{p^{\alpha}+1}=a, \ b^{p^{\beta}+1}=b \bigr\rangle, $$ where ????2??,????????1 and ??+??>3. Groups defined by these presentations for p=2 are known to play an important role in the theory of 2-groups of nilpotency class?2. We prove that |Sg(?? ₁)|=p ^??+??+?? +p ^?? +p ^?? +p ^?? +p ^??+?? +p ^??+?? +p ^??+?? and |Sg(?? ₂)|=p ^??+?? +p ^?? +p ^?? .     

Estimating asymptotic dependence functionals in multivariate regularly varying models

This paper deals with semiparametric estimation of the asymptotic portfolio risk factor ?? _?? introduced in [G. Mainik and L. Rüschendorf, On optimal portfolio diversification with respect to extreme risks, Finance Stoch., 14:593?C623, 2010] for multivariate regularly varying random vectors in $ \mathbb{R}_{+}^d $ . The functional ?? _?? depends on the spectral measure ??, the tail index ??, and the vector ?? of portfolio weights. The representation of ?? _?? is extended to characterize the portfolio loss asymptotics for random vectors in ? ^d . The earlier results on uniform strong consistency and uniform asymptotic normality of the estimates of ?? _?? are extended to the general setting, and the regularity assumptions are significantly weakened. Uniform consistency and asymptotic normality are also proved for the estimators of the functional $ \gamma_\xi^{{{1} \left/ {\alpha } \right.}} $ that characterizes the asymptotic behavior of the portfolio loss quantiles. The techniques developed here can also be applied to other dependence functionals.     

Coquasitriangular Hopf group coalgebras and braided monoidal categories

Let ?? be a group, and let H be a Hopf ??-coalgebra. We first show that the category M ^H of right ??-comodules over H is a monoidal category and there is a monoidal endofunctor (F _?? , id, id) of M ^H for any ?? ?? ??. Then we give the definition of coquasitriangular Hopf ??-coalgebras. Finally, we show that H is a coquasitriangular Hopf ??-coalgebra if and only if M ^H is a braided monoidal category and (F _?? , id, id) is a braided monoidal endofunctor of M ^H for any ?? ?? ??.     

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ³. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium).