The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integr...The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.展开更多
For a compact symplectic manifold which is s-Lefschetz which is weaker than the decomposition for de hard Lefschetz property, we prove that the Lefschetz Rham cohomology also holds.
A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S^(4n-1)→ S^(2n)× S^m as a module and demonstrate that it is not necessarily determined by the map in...A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S^(4n-1)→ S^(2n)× S^m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.展开更多
In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.
The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” Kǎihler manifolds （e.g. Kǎihler hyperbolic manifolds, bounded domains of holomorphy） by using the Bergman kerne...The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” Kǎihler manifolds （e.g. Kǎihler hyperbolic manifolds, bounded domains of holomorphy） by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.展开更多
The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partiallypositive bisectional curvature without the assumption of nonnegative bisectional curvature.Some applications of the results to ...The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partiallypositive bisectional curvature without the assumption of nonnegative bisectional curvature.Some applications of the results to holomorphic mappings are given.展开更多
For a symplectic manifold(M^(2n), ω) without boundary(not necessarily compact), we prove Poincaré type duality in filtered cohomology rings of differential forms on M, and we use this result to obtain duality be...For a symplectic manifold(M^(2n), ω) without boundary(not necessarily compact), we prove Poincaré type duality in filtered cohomology rings of differential forms on M, and we use this result to obtain duality between(d + d~Λ)-and dd~Λ-cohomologies.展开更多
Abstract. We find the minimal free resolution of a fat star-configuration X in Pn of type （r, s,t） defined by general forms of degrees d1,...,dr, and show that a fat linear star- configuration X in P2 never has gene...Abstract. We find the minimal free resolution of a fat star-configuration X in Pn of type （r, s,t） defined by general forms of degrees d1,...,dr, and show that a fat linear star- configuration X in P2 never has generic Hilbert function if （s,t） ≠ （1, 1） or （2, 2）. These two results generalize the interesting results of [2].展开更多
Let f ： M→ M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N（f） of f is equal to the minimal number of fixed points of f＇ among all self-maps f＇ in the homotopy class of f. In ...Let f ： M→ M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N（f） of f is equal to the minimal number of fixed points of f＇ among all self-maps f＇ in the homotopy class of f. In this paper, we determine N（f） for all self-maps f when M is a closed 3-manifold with S^2× R geometry. The calculation of N（f） relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.展开更多
We derive averaging formulas for the Lefschetz coincidence numbers,the Nielsen coincidence numbers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected so...We derive averaging formulas for the Lefschetz coincidence numbers,the Nielsen coincidence numbers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected solvable Lie group of type(R).As an application,we compare our formula for the Nielsen coincidence numbers with a result of Jezierski(1992)for pairs of maps on some infra-solvmanifolds of Sol.For all the pairs of self-maps of a nonorientable infra-solvmanifold of Sol,we determine the sets of all the possible values of the Nielsen coincidence numbers and the Reidemeister coincidence numbers.展开更多
文摘The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.
基金Acknowledgements The authors were very grateful to their advisor Prof. Hongyu Wang for discussion and suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11371309, 11471145, 11401514）, the University Science Research Project of Jiangsu Province （14KJB110027）, and the Foundation of Yangzhou University （2014CXJ004）.
文摘For a compact symplectic manifold which is s-Lefschetz which is weaker than the decomposition for de hard Lefschetz property, we prove that the Lefschetz Rham cohomology also holds.
基金supported by the National Science and Engineering Research Council of Canada
文摘A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S^(4n-1)→ S^(2n)× S^m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.
文摘In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.
基金supported by the Program for New Century Excellent Talents in University of China (No. 050380)the Grant for Chinese Excellent Doctorate’s Degree Thesis (No. 200519)the Fok Ying Tung EducationFoundation and the National Natural Science Foundation of China (No. 10871145).
文摘The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” Kǎihler manifolds （e.g. Kǎihler hyperbolic manifolds, bounded domains of holomorphy） by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.
文摘The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partiallypositive bisectional curvature without the assumption of nonnegative bisectional curvature.Some applications of the results to holomorphic mappings are given.
基金Supported by NSFC(Grant Nos.11521101 and 11601534)
文摘For a symplectic manifold(M^(2n), ω) without boundary(not necessarily compact), we prove Poincaré type duality in filtered cohomology rings of differential forms on M, and we use this result to obtain duality between(d + d~Λ)-and dd~Λ-cohomologies.
文摘Abstract. We find the minimal free resolution of a fat star-configuration X in Pn of type （r, s,t） defined by general forms of degrees d1,...,dr, and show that a fat linear star- configuration X in P2 never has generic Hilbert function if （s,t） ≠ （1, 1） or （2, 2）. These two results generalize the interesting results of [2].
基金supported in part by Projeto Tematico Topologia Algebrica Geometrica e Differencial2008/57607-6supported in part by NSFC(Grant No.10931005)project of Beijing Municipal Education Commission(Grant No.KZ201310028030)
文摘Let f ： M→ M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N（f） of f is equal to the minimal number of fixed points of f＇ among all self-maps f＇ in the homotopy class of f. In this paper, we determine N（f） for all self-maps f when M is a closed 3-manifold with S^2× R geometry. The calculation of N（f） relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.
基金supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(Grant No.NRF-2016R1D1A1B01006971)。
文摘We derive averaging formulas for the Lefschetz coincidence numbers,the Nielsen coincidence numbers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected solvable Lie group of type(R).As an application,we compare our formula for the Nielsen coincidence numbers with a result of Jezierski(1992)for pairs of maps on some infra-solvmanifolds of Sol.For all the pairs of self-maps of a nonorientable infra-solvmanifold of Sol,we determine the sets of all the possible values of the Nielsen coincidence numbers and the Reidemeister coincidence numbers.
