Finite Transformation Groups Research Articles

A Note on Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

Applying the classical Lie symmetry approach to the barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a β-plane channel, we find a new symmetry, which is not presented in a previous work [F. Huang, Commun. Theor. Phys. (Beijing, China) 42 (2004) 903]. A general finite transformation group is obtained based on the full Lie point symmetry. Furthermore, two new types of similarity reduction solutions are obtained.

Lie point symmetry algebras and finite transformation groups of the general Broer–Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer–Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.

ULTRACOHERENCE AND CANONICAL TRANSFORMATIONS

The (in)finite dimensional symplectic group of homogeneous canonical transformations is represented on the bosonic Fock space by the action of the group on the ultracoherent vectors, which are generalizations of the coherent states.

Finite symmetry transformation groups and exact solutions of Lax integrable systems

Abstract In this paper, a method is developed to directly find finite symmetry transformation groups and then symmetries of Lax integrable nonlinear physical systems. Some symmetry groups, symmetry algebras and exact solutions for the Kadomtsev–Petviashvili equation, the dispersive long-wave equation and the extended self-dual Yang–Mills equations are explicitly given.

Finite symmetry transformation groups and exact solutions of Lax integrable systems

In this paper, the finite symmetry transformation groups and then symmetries of Lax integrable nonlinear physical systems, the Davey–Stewartson equation and the (2+1)-dimensional Camassa–Holm equation are investigated by means of a simple direct method.

Equations of arbitrary order invariant under the Kadomtsev–Petviashvili symmetry group

By means of a simple new approach, a general Kadomtsev–Petviashvili (KP) family with an arbitrary function of group invariants of arbitrary order is proposed. It is proved that the general KP family possesses a common infinite dimensional Kac–Moody–Virasoro Lie point symmetry algebra. The known fourth order one can be re-obtained as a special example. The finite transformation group is presented in a clearer form. The Kac–Moody–Virasoro group invariant solutions and the Kac–Moody group invariant solutions of the KP family are determined by the Boussinesq and KdV families, respectively.

Symmetries and conserved quantities for systems of generalized classical mechanics

In this paper, the symmetries and the conserved quantities for systems of generalized classical mechanics are studied. First, the generalized Noether’s theorem and the generalized Noether’s inverse theorem of the systems are given, which are based upon the invariant properties of the canonical action with respect to the action of the infinitesimal transformation of r-parameter finite group of transformation; second, the Lie symmetries and conserved quantities of the systems are studied in accordance with the Lie’s theory of the invariance of differential equations under the transformation of infinitesimal groups; and finally, the inner connection between the two kinds of symmetries of systems is discussed. PACC: 0320; 1110; 0220 I. INTRODUCTION The conservation laws (or first integrals) of the mechanical systems are always of mathematical importance and at the same time, they are regarded as the manifestation of some profound physical principles. The fact that the existence of a conserved quantity means that an inner dynamical symmetry in the mechanical system plays an irreplaceable role in the physical interpretation of motion. The modern methods in finding conservation laws are mainly two ways [1] with each having a concept of its own: the first one is based upon the invariant properties of Hamiltonian action with respect to the action of the infinitesimal transformation of the finite transformation group, i.e., Noether symmetries and conserved quantities; the second one is based upon the Lie’s theory of the invariance of differential equations under the action of infinitesimal groups. The differential equations of many physical problems appear to be the variational problems. The theory of the generalized classical mechanics, initiated by Ostrogradsky and Jacobi in the years from 1848 to 1858, has been gaining much development during the recent fifty years, and many important results have been made. [2 6] The Lagrange equations with high

Simple Diagonal Locally Finite Lie Algebras

The ground field F is an algebraically closed field of zero characteristic, and N is the set of natural numbers. Let L be a locally finite-dimensional (or locally finite, for brevity) Lie algebra. Assume for simplicity that L has countable dimension. Then one can choose finite-dimensional subalgebras Li i ∈ N of L in such a way that Li ⊆ Li+1 for all i and L = ∪i∈NLi. The set Li | i ∈ N is called a local system of L. Assume that L is simple. Then all Li can be chosen to be perfect (that is, [Li, Li] = Li). We shall call such local systems perfect. Let S i = S i 1 ⊕ … ⊕ S i n i be a Levi subalgebra of Li, where S i 1 , … , S i n i are simple components of Si, and let V i k be the standard S i k -module. Since Li is perfect, for every k there exists a unique irreducible Li-module V i k such that the restriction V i k ↓ S i k is isomorphic to V i k . An embedding Li → Li+1 is called diagonal if, for each l, any non-trivial composition factor of the restriction \calV i + 1 l ↓ L i is isomorphic to V i k or its dual V i k ∗ for some k. We prove that a simple locally finite Lie algebra L can be embedded into a locally finite associative algebra if and only if there exists a perfect local system { L i } i ∈ N of L such that all embeddings Li → Li+1 are diagonal. Note that this result can be considered as a version of Ado's theorem for simple locally finite Lie algebras. Let V be a vector space. An element x ∈ fgl(V) is called finitary if dim xV < ∞. The finitary transformations of V form an ideal fgl(V) of gl(V), and any subalgebra of fgl(V) is called a finitary Lie algebra. We classify all finitary simple Lie algebras of countable dimension. It turns out that there are only three: sl∞(F), so∞(F), and sp∞(F). The author has classified all finitary simple Lie algebras and will describe this in a subsequent paper. The analogous problem in group theory has been recently solved by J.I. Hall. He classified simple locally finite groups of finitary linear transformations. 1991 Mathematics Subject Classification: 17B35, 17B65.

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic.

Heteroclinic period blow-up in certain symmetric ordinary differential equations

We consider the problem of bifurcation as well as of accumulation of periodic orbits on heteroclinic orbits for certain systems of ordinary differential equations either equivariant under finite groups of linear transformations or periodic in spatial variables.

On the Kummer Solutions of the Hypergeometric Equation

One of the oldest, and still one of the most interesting, applications of group theory arises in the study of the transformations of an ordinary differential equation. If we know that a given differential equation admits a group of transformations, then we know that the solution set must admit that same group of transformations, and we can deduce properties of all the solutions from the properties of any one of them. A case in point is offered by the celebrated hypergeometric equation (See Eq. (1) below), whose solutions include many of the most interesting special functions of mathematical physics. In his book [3], Einar Hille notes that this equation has a venerable history associated with such names as Gauss, Euler, Riemann, and Kummer. The hypergeometric equation is in fact a prototype: every ordinary differential equation of second order with at most three regular singular points can be brought to the hypergeometric equation by means of suitably chosen changes of variable [S]. In 1836 Kummer published a set of 6 distinct solutions of the hypergeometric equation. These include the hypergeometric function of Gauss, and all of them could be expressed in terms of Gauss's function (See Table 1 below). A useful summary of their basic properties is found in [1, p. 105ff.]. A glance at the list of these Kummer solutions reveals a rather complicated set of relationships which pleads for some simple explanation. We show here that the Kummer solutions are related by a finite group of transformations which serve to explain their relationships and to exemplify the use of transformation groups in the study of differential equations.

A p�lya urn model with a continuum of colors

For a Polya urn model with a continuum of colors introduced by Blackwell and MacQueen ((1973),Ann. Statist.,2, 1152–1174), we show the joint distribution of colors aftern draws from which several properties of the urn model are derived. The similar results hold for the case where the initial distribution of colors is invariant under a finite group of transformations.

On representations of finite transformation groups of algebraic curves and surfaces

We study the G-representations H ∗( X; F) arising from G-actions on algebraic varieties X and G-sheaves F in terms of the geometry of the action.

Integral representations of finite transformation groups III: simply-connected four-manifolds

The action of a finite group G on a simply-connected 4-manifold X yields a Z G-lattice H 2( X). In this paper we study the relations between the geometry of the G-action on X and the homological and representation theoretic properties of H 2( X).

Integral representations of finite transformation groups I

Soit X un G-espace ou G est un groupe fini. Les groupes d'homologie et de cohomologie H * (X;R) et H * (X;R) admettent une structure induite de RG-module, ou R est un anneau commutatif muni de l'identite. On etudie les proprietes homologiques de tels modules

Integral representations of finite transformation groups II: non-simply-connected spaces

Introduction Let r be a discrete group acting on a topological space X, or more generally on a pair of spaces (Y, X). The induced action on homology gives Ha(X; R) (respective- ly H,( Y, X; R)) an RT-module structure. Question. What type of restrictions does the topology of the underlying space im- pose on the r-module structure of these modules? We study this problem from the point of view of homological algebra in certain circumstances motivated by specific topological questions. More specifically, we show that RT-projectivity of H,( Y, X; R) and H*(X; R) is detected by restriction to ‘virtually cyclic’ subgroups of r. This result generalizes our earlier theorems in [5,6] for the case Irl< 00 to the effect that RT-projectivity of H*( Y, X; R) and H*(X; R) is detected by restriction to prime order subgroups of r. Let (YX) be a pair of connected ‘reasonable’ topological spaces (e.g. CW com- plexes etc.) with rri(X)~ ~cr( Y)g n, and let G be a finite group acting on (Y,X). Suppose

Estimation under invariant distributions

If a distribution is invariant under a finite group of transformations, an estimator of a parameter associated with the distribution is improved by making it invariant. The resulting invariant estimator is characterized as the projection of the original estimator. If the estimator is the uniformly minimum variance unbiased estimator of its expectation for continuous distributions, then the invariant estimator is the uniformly minimum variance unbiased estimator for invariant and continuous distributions. A typical example is U-statistics and invariant U-statistics.

Varieties in finite transformation groups

The equivariant cohomology ring of a G-space X defines a homogeneous affine variety. Quillen [Q] and W. Y. Hsiang [Hs] have determined the relation between such varieties and the family of isotropy subgroups as well as their fixed point sets when dim X < oo. In modular representation theory, J. Carlson [Cj] introduced cohomological support varieties and rank varieties (the latter depending on the group algebra) and explored their relationship. We define rank and support varieties for G-spaces and G-chain complexes and apply them to cohomological problems in transformation groups. As a corollary, a useful criterion for ZG-projectivity of the reduced total homology of certain G-spaces is obtained, which improves the projectivity criteria of Rim [R], Chouinard [Ch], and Dade [D].

Some local-global results in finite transformation groups

Some local-global results in finite transformation groups

Optimal spherical designs and numerical integration on the sphere

So-called spherical t-designs in three-dimensional Euclidean space R 3 are considered. It is shown that the maximal possible value of t for designs which are permuted transitively under a finite group of Euclidean transformations is 9. This means that we can find a set of (60) points on the surface of the unit sphere in R 3 which form an orbit under the icosahedral group such that the average of the function values at the 60 points and the average over the spherical surface are identical for all spherical harmonics of degree less than or equal to t = 9, while there is no such orbit for t = 10. Among all 9-designs which are orbits of the icosahedral group exactly one is optimal with respect to approximation of integrals of harmonics of tenth and eleventh order. This set is given in Section 2 of this paper. It is very well suited for numerically evaluating integrals of continuous functions over the sphere, a problem occurring very often in mathematics and science, as, e.g., physics, astronomy, etc.