The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Group theoretical modeling of thermal explosion with reactant consumption

Today engineering and science researchers routinely confront problems in mathematical modeling involving nonlinear differential equations. Many mathematical models formulated in terms of nonlinear differential equations can be successfully treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of the symmetrical heating of an exothermally reacting medium with approximations to the body’s temperature distribution similar to those made by Thomas [17] and Squire [15]. The quantitative results were found to be in a good agreement with Adler and Enig in [1], where the authors were comparing the integral curves corresponding to the critical conditions for the first-order reaction. Further development of the modeling by including the critical temperature is proposed. Overall, it is shown, in particular, that the application of Lie group analysis allows one to extend the previous analytic results for the first order reactions to nth order ones.     

可允许度量下有限能量辛涡旋的渐近行为献给钱敏教授90华诞

假设(X,ω)是一个具有紧致单连通Lie群G Hamilton作用的紧致光滑辛流形.本文证明只要Riemann面的柱形端口具有一个比标准柱形度量增长速度快的线性度量,那么任何一个有限能量辛涡旋将以指数衰减的速度收敛到辛流形X在正则值辛约化的扭曲分支或非扭曲分支上.本文结果无需假设群G在正则水平集上的作用是自由的.因此,它直接推广了Ziltener在群作用自由的假设下得出的相关结果.本文结果在作者关于量子化Kirwan同态的系列工作中有重要应用.     

Lifting smooth curves over invariants for representations of compact Lie groups

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.P. W. M. was supported by Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 10037 PHY.     

Lie symmetries and the center problem

In this short survey we study the narrow relation between the center problem and the Lie symmetries. It is well known that an analytic vector eld X having a non-degenerate center has a non-trivial analytic Lie symmetry in a neighborhood of it, i.e. there exists an analytic vector eld Y such that [X;Y] = \(\mu\)X. The same happens for a nilpotent center with an analytic rst integral as can be seen from the last results about nilpotent centers. From the last results for nilpotent and degenerate centers it also can be proved that any nilpotent or degenerate center has a trivial smooth (of class \(C^{\infty} \) ) ) Lie symmetry. Remains open if always exists also a non-trivial Lie symmetry for any nilpotent and degenerate center.     

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820-839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.     

Noetherianness of wreath products of Abelian Lie algebras with respect to equations of universal enveloping algebra

It is proved that a wreath product of two Abelian finite-dimensional Lie algebras over a field of characteristic zero is Noetherian w.r.t. equations of a universal enveloping algebra. This implies that an index 2 soluble free Lie algebra of finite rank, too, has this property.     

Minimizing coincidence in positive codimension

Let f and g be maps between smooth manifolds M and N of dimensions n + m and n, respectively (where m > 0 and n > 2). Suppose that the image (fxg)(M) intersects the diagonal N × N in finitely many points, whose preimages are smooth m-submanifolds inM. The problem of minimizing the coincidence set Coin(f, g) of the maps f and g with respect to these preimages and/or their components is considered. The author’s earlier results are strengthened. Namely, sufficient conditions under which such a coincidence m-submanifold can be removed without additional dimensional constraints are obtained.     

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

The spherical functions on a real semisimple Lie group (w.r.t. a maximal compact subgroup) are characterized as joint eigenfunctions of certain differential operators on the corresponding complex group. Using this, several results concerning the spherical Fourier transform on the real group are reduced to the corresponding results for the complex group.When the group in question is a normal real form, this leads to new and simpler proofs of such results as the Plancherel formula, the Paley-Wiener theorem and the characterization of the image under the spherical Fourier transform of the L¹- and L²-Schwartz spaces. In these proofs neither any knowledge of Harish-Chandras c-function nor the series expansion for the spherical function are used.For the proof of the main result some analysis of independent interest on pseudo-Riemannian symmetric spaces is developed. Such as a generalized Cartan decomposition and a method of analytic continuation between two “dual” pseudo-Riemannian symmetric spaces.     

A connection-theoretic approach to reduction of second-order dynamical systems with symmetry

We deal with reduction of Lagrangian systems that are invariant under the action of the symmetry group. Unlike the bulk of the literature we do not rely on methods coming from the calculus of variations. Our method is based on the geometrical analysis of regular Lagrangian systems, where solutions of the Euler-Lagrange equations are interpreted as integral curves of the associated second-order differential equation field. In particular, we explain so-called Lagrange-Poincaré reduction [1] and Routh reduction [3] from the viewpoint of that vector field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetry reductions of the Lax pair for the 2 + 1-dimensional Konopelchenko–Dubrovsky equation

Based on the symbolic computational system – Maple, the similarity reduction arising from the classical Lie point symmetries of the Lax pair for the 2 + 1-dimensional Konopelchenko–Dubrovsky (KD) equation, is carried out. We obtain several interesting reductions. By analyzing not only the reduced Lax pair but also the KD equation reduced under the same symmetry group, we find that the reduced Lax pairs do not always lead to the reduced KD equation.     

Anosov diffeomorphisms on infra-nilmanifolds associated to graphs

Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore, their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, that is, a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this question for the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled on free nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifolds associated to graphs, leading to a necessary and sufficient condition depending only on the induced action of the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of the holonomy group on simple graphs.     

Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

The Lie Group Structure of the Butcher Group

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge–Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker–Campbell–Hausdorff Lie group modelled on a Fréchet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated with the Butcher group by Connes and Kreimer.     

Polygon Recognition and Symmetry Detection

We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for the recognition of polygons in R² modulo the special Euclidean, Euclidean, equi-affine, skewed-affine, and similarity Lie groups. We also solve the case of polygons in the Poincar\'e half-plane under the action of SL(2) and explain a method applicable to Lie group actions in general. The time complexity of our algorithms is linear in the number of vertices and they are noise resistant. The signatures used allow the detection of partial, as well as approximate, equivalences.     

Control of mechanical systems on Lie groups and ideal hydrodynamics

In contrast to the Euler–Poincaré reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is the velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Now we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant with respect to left translations on this group, and assume that the mass geometry f the system may change under the action of internal control forces. Such a system can also be reduced to a Lie group. Without controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and, therefore, its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Derivation of conservation laws for a nonlinear wave equation modelling melt migration using Lie point symmetry generators

The derivation of conservation laws for a nonlinear wave equation modelling the migration of melt through the Earth’s mantle is considered. New conserved vectors which depend explicitly on the spatial coordinate are generated using the Lie point symmetry generators of the equation and known conserved vectors. It is demonstrated how conserved vectors that are conformally associated with a Lie point symmetry generator can be derived more simply than by the direct method by imposing the symmetry condition on the conservation law equation.     

Riemannian metrics on Teichmüller space

On each compact Riemann surface Σ of genusp≥1, we have the Bergman metric obtained by pulling back the flat metric on its Jacobian via the Albanese map. Taking theL ²-product of holomorphic quadratic differentials w.r.t. this metric induces a Riemannian metric on the Teichmüller spaceT _p that is invariant under the action of the modular group. We investigate geometric properties of this metric as an alternative to the usually employed Weil-Petersson metric. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.     

Analytic dependence of riemann mappings for bounded domains and minimal surfaces

In this paper, we prove that the functional which takes a closed Lavrentiev curve to the corresponding Riemann mapping is locally Lipl on the set Ω of all closed Lavrentiev curves. This set is a subset of BMO(T). It is, however, not open in BMO(T). We also prove that the previous functional is analytic for certain classes of closed Lavrentiev curves, including the class of curves which have some symmetry with respect to the unit circle. These classes of curves are submanifolds of BMO(T). Finally, we consider the functional which takes a Lavrentiev curve (closed or not) in n-dimensional Euclidean space to the corresponding minimal surface, and we study the differentiability and analyticity of this functional on certain function spaces. © 1993 John Wiley & Sons, Inc.