As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the ...As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on R<sup>n</sup>. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.展开更多
There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M →4 M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g continuous} and NJDn（f） = ...There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M →4 M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g continuous} and NJDn（f） = min{#Fix（gn）;g - f; g smooth}. In general, NJDn（f） may be much greater than NFn（f）. We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NFn（f） = NJDn（f） holds for all n →← all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.展开更多
文摘As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on R<sup>n</sup>. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.
基金supported by the National Science Center,Poland(Grant No.UMO2014/15/B/ST1/01710)
文摘There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M →4 M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g continuous} and NJDn（f） = min{#Fix（gn）;g - f; g smooth}. In general, NJDn（f） may be much greater than NFn（f）. We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NFn（f） = NJDn（f） holds for all n →← all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.
