The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics...The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kiihler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider arid deeper results. differential展开更多
Let F_(q)be a finite field of any characteristic and GL(n,F_(q))be the general linear group over F_(q).Suppose W denotes the standard representation of GL(n,F_(q)),and GL(n,F_(q))acts diagonally on the direct sum of W...Let F_(q)be a finite field of any characteristic and GL(n,F_(q))be the general linear group over F_(q).Suppose W denotes the standard representation of GL(n,F_(q)),and GL(n,F_(q))acts diagonally on the direct sum of W and its dual space W^(∗).Let G be any subgroup of GL(n,F_(q)).Suppose the invariant field F_(q)(W)G=F_(q)(f1,f2,…,fk),where f1,f2,…,fk in F_(q)[W]G are homogeneous invariant polynomials.We prove that there exist homogeneous polynomialsl1,l2,…,ln in the invariant ring F_(q)[W⊕W^(∗)]G such that the invariant field F_(q)(W⊕W^(∗))G is generated by{f1,f2,…,fk,l1,l2,…,ln}over F_(q).展开更多
Given a system of vector fields on a smooth manifold that spans a plane field of constant rank, we present a systematic method and an algorithm to find submanifolds that are invariant under the flows of the vector fie...Given a system of vector fields on a smooth manifold that spans a plane field of constant rank, we present a systematic method and an algorithm to find submanifolds that are invariant under the flows of the vector fields. We present examples of partition into invariant submanifolds, which further gives partition into orbits. We use the method of generalized Frobenius theorem by means of exterior differential systems.展开更多
This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was...This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as <i>The General Linear Group GL(n, R)</i>, The <i>Complex General Linear Group GL(n, C)</i>, <i>The Special Linear Group SL(n, R)=SL(V)</i>, <i>The Complex Special Linear Group SL(n, C)</i>, <i>Unitary and Orthogonal Groups</i>, <i>Symplectic Group</i>, <i>The groups R*, C*, S<sup>1</sup> and R<sup>n</sup></i> and others. In addition, invariant vector fields and the exponential map and the lie algebra of a lie group. For me, this work has the significance of being useful to all who need it.展开更多
文摘The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kiihler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider arid deeper results. differential
基金This research was partially supported by the NNSF of China(No.11301061).
文摘Let F_(q)be a finite field of any characteristic and GL(n,F_(q))be the general linear group over F_(q).Suppose W denotes the standard representation of GL(n,F_(q)),and GL(n,F_(q))acts diagonally on the direct sum of W and its dual space W^(∗).Let G be any subgroup of GL(n,F_(q)).Suppose the invariant field F_(q)(W)G=F_(q)(f1,f2,…,fk),where f1,f2,…,fk in F_(q)[W]G are homogeneous invariant polynomials.We prove that there exist homogeneous polynomialsl1,l2,…,ln in the invariant ring F_(q)[W⊕W^(∗)]G such that the invariant field F_(q)(W⊕W^(∗))G is generated by{f1,f2,…,fk,l1,l2,…,ln}over F_(q).
基金supported by National Research Foundation of Republic of Korea (Grant No. 2011-0008976)
文摘Given a system of vector fields on a smooth manifold that spans a plane field of constant rank, we present a systematic method and an algorithm to find submanifolds that are invariant under the flows of the vector fields. We present examples of partition into invariant submanifolds, which further gives partition into orbits. We use the method of generalized Frobenius theorem by means of exterior differential systems.
文摘This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as <i>The General Linear Group GL(n, R)</i>, The <i>Complex General Linear Group GL(n, C)</i>, <i>The Special Linear Group SL(n, R)=SL(V)</i>, <i>The Complex Special Linear Group SL(n, C)</i>, <i>Unitary and Orthogonal Groups</i>, <i>Symplectic Group</i>, <i>The groups R*, C*, S<sup>1</sup> and R<sup>n</sup></i> and others. In addition, invariant vector fields and the exponential map and the lie algebra of a lie group. For me, this work has the significance of being useful to all who need it.
