Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: autonomous systems

This paper is concerned with the symbolic computation of the normal forms of general multiple-degree-of-freedom oscillating systems. A perturbation technique based on the method of multiple time scales, without the application of center manifold theory, is generalized to develop efficient algorithms for systematically computing normal forms up to any high order. The equivalence between the perturbation technique and Poincaré normal form theory is proved, and general solution forms are established for solving ordered perturbation equations. A number of cases are considered, including the non-resonance, general resonance, resonant case containing 1:1 primary resonance, and combination of resonance with non-resonance. “Automatic” Maple programs have been developed which can be executed by a user without knowing computer algebra and Maple. Examples are presented to show the efficiency of the perturbation technique and the convenience of symbolic computation. This paper is focused on autonomous systems, and non-autonomous systems are considered in a companion paper.     

Parametrized Gröbner–Shirshov bases

We consider the problem of describing Gröbner–Shirshov bases for free associative algebras in finite terms. To this end we consider parametrized elements of an algebra and give methods for working with them which under favorable conditions lead to a basis given by finitely many patterns. On the negative side we show that in general there can be no algorithm. We relate our study to the problem of verifying that a given set of words in certain groups yields Bokut’ normal forms (or groups with a standard basis).     

拟线性微分代数方程的标准形

讨论了拟线性微分代数方程在一类特殊的奇点-拟障碍点附近的标准形.通过矩阵广义逆理论,拟线性微分代数方程可化为半显式形式.然后运用标准形理论,在微分同胚变换下,给出了拟线性微分代数方程在拟障碍点附近的标准形.在此基础上进一步讨论了这类标准形的去奇异化性质.     

The recognition of a class of bifurcation problems

In bifurcation theory there are two recognition problems concerning a given normal form, the recognition for the normal form and the recognition for universal unfoldings of bifurcation problems which are equivalent to the normal form. The two recognition problems for the normal forms εx²+δλ^k were only partially solved. In this paper we give a complete solution of the two problems for all k?1 uniformly.     

The Lie Algebra of a Hyperbolic Higher Degree Alternating Form

We compute the Lie algebra of a higher degree hyperbolic alternating form, which generalizes the symplectic algebra to higher degrees. We use this to show that hyperbolic forms descend under scalar extension. An immediate consequence is that a version of the weak Hasse–Minkowski Theorem is valid for alternating forms of higher degree.     

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above. In this paper, we generalize the theory to p.v.’s with symmetric structure ofK _ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. Dedicated to the memory of Professor K G Ramanathan     

环上的Clifford代数

§1. 引言与记号 如众周知,域上的Clifford代数乃是概括域上的Grassmann代数(外代数)以及广义四元数代数的一个代数。它不但在数学的一些分支(如群表示论、二次型理论等)中有着重要的应用,而且也是近代理论物理中的有用工具之一(比如参看[1])。1954年,C.Chevalley在[2]中完美地给出了域上Clifford代数的基本理论。本文的主要目的是建立可换环上的Clifford代数,即给出它的定义、存在性与唯一性等。容易看出,这是域上的Clifford代     

Quadratic descent of hermitian forms

Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.     

Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.     

Quadratic forms in skew normal variates

In this paper first a characterization of the multivariate skew normal distribution is given. Then the joint moment generating functions of two quadratic forms, and a linear compound and a quadratic form in skew normal variates, have been derived and conditions for their independence are given. Distribution of the ratios of quadratic forms in skew normal variates has also been studied.     

Generalized Exponents and Forms

We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a linear character) using the theory of semi-invariant differential forms. Springer’s theory of regular numbers gives a formula when the group is generated by dim V reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.     

Diagonalization and Jordan normal form – motivation through Maple ®

Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package Maple®.     

Note on Co-split Lie Algebras

It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.     

Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004     

Generalized modular forms

The theory of “generalized modular forms,” initiated here, grows naturally out of questions inherent in rational conformal field theory. The latter physical theory studies q-series arising as trace functions (or partition functions), which generate a finite-dimensional SL(2,Z)-module. It is a natural step to investigate whether these q-series are in fact modular forms in the classical sense. As it turns out, the existence of the module does not, of itself, guarantee that this is so. Indeed, our Theorem 1 shows that such q-series of necessity behave like modular forms in every respect, with the important exception that the multiplier system need not be of absolute value one. The Supplement to Theorem 1 shows that such q-series are classical modular forms exactly when the scalars relating the q-series generators of the module have absolute value one. That is, the SL(2,Z)-module in question is unitary. (There is the further restriction that the associated representation is monomial.) We prove as well that there exist generalized modular forms which are not classical modular forms. (Hence, as asserted above, the q-series need not be classical modular forms.)Beyond Theorem 1 and its Supplement, which serve to relate our generalized modular forms to classical modular forms (and thus justify the name), this work develops a number of their fundamental properties. Among these are a basic result relating generalized modular forms to classical modular forms of weight 2 and so, as well, to abelian integrals. Further, we prove two general existence results and a complete characterization of weight k generalized modular forms in terms of generalized modular forms of weight 0 and classical modular forms of weight k.     

Biderivations,linear commuting maps and commutative post-Lie algebra structures on W-algebras

In this paper, the biderivations without the skew-symmetric condition of W-algebras including the Witt algebra, the algebra W(2,2) and their central extensions are characterized. Some classes of non-inner biderivations are presented. As applications, the forms of linear commuting maps and the commutative post-Lie algebra structures on aforementioned W-algebras are given.     

Poisson Brackets of Even Symplectic Forms on the Algebra of Differential Forms

Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established.     

Invariant symmetric bilinear forms for reflection groups

In this paper we describe a connection between Vinberg's criterion for the existence of an invariant symmetric bilinear form for a geometric representation of a Coxeter groups and other criteria which are formulated in terms of conjugation invariant sets of reflections generating a given group. Similar methods lead to the result that every non-symmetrizable Kac--Moody Lie algebra contains a non-symmetrizable subalgebra of rank 3. Finally we explain how the results for symmetric bilinear forms can also be obtained for skew-symmetric forms. Received 3 March 2000.