The paper deals with rational maps between real algebraic sets. We are interested in the rational maps which extend to continuous maps defined on the entire source space. In particular, we prove that every continuous map between unit spheres is homotopic to a rational map of such a type. We also establish connections with algebraic cycles and vector bundles.

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Rational maps in real algebraic geometry

On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we get some inequalities of various total scalar curvatures and some characterization results of the Kahler metric, balanced metric, locally conformally Kahler metric and the k-Gauduchon metric. As corollaries, we show some results related to a problem given by Lejmi-Upmeier \cite{LeU} and a conjecture given by Angella-Otal-Ugarte-Villacampa \cite{AOUV}.

Scalar curvatures in almost Hermitian geometry and some applications

By using moving frame theory, we obtain some necessary conditions involving curvatures for integrability of an almost Hermitian structure. As consequences, they are applied to S6.

Curvature and integrability of an almost hermitian structure

Considering the multi-sources, heterogeneity and complexity of dam safety assessment, a dam safety assessment model based on interval-valued intuitionistic fuzzy set and evidence theory is proposed to perform dam safety reliability evaluations. In the proposed model, the dynamic reliability based on the supporting degree is applied to modify the data from homologous information. The interval-valued intuitionistic fuzzy set is used to describing the uncertainty and fuzziness between heterogeneous information. Evidence theory is employed to integrate the data from heterogeneous information. Finally, a multiple-arch dam undergoing structural reinforcement is taken as an example. The evaluation result before reinforcement shows that the safety degree of the dam is low and the potential risk is more likely to be located at the dam section #13. From the geological survey before reinforcement, there exist weak fracture zone and broken mud belt in the foundation of the dam section #13. The comparison between the evaluation results before and after reinforcement indicates that the dam become safer and more stable after reinforcement.

Dam Safety Evaluation Based on Interval-Valued Intuitionistic Fuzzy Sets and Evidence Theory

Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and $$X \subseteq V$$
 some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, a continuous map $$f :X \rightarrow W$$
 is said to be piecewise-regular if there exists a stratification $$\mathscr {S}$$
 of V such that for every stratum $$S \in \mathscr {S}$$
 the restriction of f to each connected component of $$X \cap S$$
 is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V. Assuming that the subset X of V is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact and nonsingular, we prove that each continuous map from V into a unit sphere is homotopic to a piecewise-regular map of class $$\mathscr {C}^k$$
 , where k is an arbitrary nonnegative integer.

/pdf/piecewise-regular-maps-y30oitb3ya.pdf

Piecewise-regular maps

Some homotopy classes of the unit spheres are explicitly represented by entire rational maps.

Algebraic maps from spheres to spheres

Over an oriented even dimensional Riemannian manifold(M
 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm
 = SO(2m)/U(m)↪
 J
 +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ
 1 onJ
 +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.

Zizhou Tang

Papers

Algebraic maps from spheres to spheres

Twistor bundle theory and its application