We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, whi...We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, which we used to simulate the coupling constant hierarchy for the universe with five fundamental interactions. Moreover, we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph. This matrix coincides up to sign with the integer linking matrix (the main topological invariant) of the graph manifold corresponding to the plumbing graph. The need for a special algorithm appeared during computations of these topological invariants of complicated graph manifolds since there emerged a set of special rational numbers (fractions) with huge numerators and denominators;for these rational numbers, the ordinary methods of expansion in continued fraction became unusable.展开更多
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 inv...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<sup>2</sup> or in CP<sup>2</sup>.In this article,we show that these groups,for the Hirzebruch surface F<sub>1</sub>,（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
Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a c...Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a closed connected smooth manifold. We say that M has an almost structure if its tangent bundle has a complex structure.展开更多
In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a c...In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.展开更多
文摘We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, which we used to simulate the coupling constant hierarchy for the universe with five fundamental interactions. Moreover, we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph. This matrix coincides up to sign with the integer linking matrix (the main topological invariant) of the graph manifold corresponding to the plumbing graph. The need for a special algorithm appeared during computations of these topological invariants of complicated graph manifolds since there emerged a set of special rational numbers (fractions) with huge numerators and denominators;for these rational numbers, the ordinary methods of expansion in continued fraction became unusable.
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<sup>2</sup> or in CP<sup>2</sup>.In this article,we show that these groups,for the Hirzebruch surface F<sub>1</sub>,（a,b）,are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.
文摘Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a closed connected smooth manifold. We say that M has an almost structure if its tangent bundle has a complex structure.
基金Supported by National Natural Science Foundation of China(Grant No.11271062)Program for New Century Excellent Talents in University(Grant No.13-0721)
文摘In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.
