Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invar...Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C^2 or in CP^2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.展开更多
Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is ...Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.展开更多
In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smoo...In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.展开更多
基金This work was supported by the Emmy Noether Institute Fellowship(by the Minerva Foundation of Germany)Israel Science Foundation(Grant No.8008/02-3)
文摘Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C^2 or in CP^2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.
基金supported by DAADEU-network HPRN-CT-2009-00099(EAGER)+2 种基金The Emmy Noether Research Institute for Mathematicsthe Minerva Foundation of GermanyThe Israel Science Foun dation grant #8008/02-3 (Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties")
文摘Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.
基金Supported by Fundamental Research Funds for the Central Universities(Key Program)National Natural Science Foundation of China(Grant No.41076125)+1 种基金973 project(Grant No.2010CB950504)Polar Climate and Environment Key Laboratory
文摘In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.
