Torsion-free path geometries and integrable second order ODE systems

A search for invariants of second order ODE systems under the class of point transformations, which mix the parameter and the dependent variables, uncovers a torsion tensor generalizing part of the curvature tensor of an affine connection. We study the geometry of ODE systems for which this torsion vanishes. These are the ODE systems for which deformations of solutions fixing a point constitute a field of Segré varieties in the tangent bundle of the locally defined space of solutions. Conversely, a field of Segré varieties for which certain differential invariants vanish induces a torsion-free ODE system on the space of solutions to a natural PDE system. The geometry on the solution space is used to produce first integrals for torsion-free ODE systems, given as algebraic invariants of a curvature tensor involving up to fourth derivatives of the equations. In the generic case, there are enough first integrals to solve the equations explicitly in spite of the absence of symmetry. In the case of torsion-free ODE pairs, the field of Segré varieties is equivalent to a half-flat split signature conformal structure, and we characterize in terms of curvature those systems having an abundance of totally geodesic surfaces.     

On Computer-Aided Solving Differential Equations and Studying Stability of Markets

For any nonholonomic manifold, i.e., a manifold with nonintegrable distribution, we define an analog of the Riemann curvature tensor and refer to Grozman's package SuperLie with the help of which the tensor had been computed in several cases. Being an analog of the usual curvature tensor, this invariant characterizes (in)stability of any nonholonomic dynamical system, in particular, of markets. Similar invariants give criteria for formal integrability of differential equations whose symmetries are induced by contact transformations similar to Goldshmidt's criteria for formal integrability of differential equations whose symmetries are induced by point transformations. As a byproduct, we obtain an approximate solution of the equation whose integrability is under study. Bibliography: 47 titles. __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 312, 2004, pp. 165–187.     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

Scalar differential invariants of symplectic Monge-Ampère equations

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form u _xy + f(x, y, u _x , u _y ) = 0 and in particular find a simple linearization criterion.     

The Euler-Jacobi-Lie integrability theorem

This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n ? 2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.     

Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians

In recent work, the authors show the mathematical structure behind both the Euler–Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. In this paper, the authors demonstrate that the knowledge of this structure allows to find the first integrals of the Euler–Lagrange equations, and subsequently, to solve by quadratures, variational problems that are invariant under the special Euclidean groups SE(2) and SE(3) .     

Differential geometry on almost tangent manifolds

Summary We start from a tensor field Q of type (1, 1) defined in a2n-dimensional manifold M which satisfies Q ²⁼⁰ and has rank n. The tensor field Q defines an almost tangent structure in M. We then introduce another tensor field P of the same type and having properties similar to those of Q. We then define and study the tensors H=PQ, V=QP, J=P−Q, K=P+Q, L=PQ−QP, (J, K, L) defining an almost quaternion structure of the second kind on M. We study the differential geometry on almost tangent manifolds in terms of these tensors. To ProfessorBeniamino Segre on his seventieth birthday Entrata in Redazione il 7 giugno 1973.     

Algebras and differential equations with additive symmetry

The set of all m-ary algebra structures on a given vector space affords, by the change of basis action, a representation of the general linear group. The invariants of a given subgroup are identified with those algebras admitting that subgroup as algebra automorphisms. Any finite dimensional representation of the additive group as automorphisms is obtained as the exponential of a nilpotent derivation. The latter can be embedded in the Lie algebra sl(2) so that the maximal vectors in an irreducible decomposition of the set of algebras as an sl(2) module are the invariants of the given action of the additive group. Dimension formulas and explicit bases are computed for the space of algebras with certain additive group actions. Employing the equivalence of the categories of m-ary algebras and systems of autonomous mth order homogeneous differential equations, the algebraic results are connected to the construction of first integrals and semi-invariants.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

Equivalence of Third-Order Ordinary Differential Equations to Chazy Equations I–XIII

In the paper we solve the equivalence problem of the third-order ordinary differential equations quadratic in the second-order derivative. For this class of equations the invariants of the group of point equivalence transformations and the invariant differentiation operators are constructed. Using these results the invariants of 13 Chazy equations were calculated. We provide examples of finding equivalent equations by use of their invariants. Also two new examples of the equations linearizable by a local transformation are found. These are a particular case of Chazy–XII equation and a Schwarzian equation.     

On Moving Frames and Noether’s Conservation Laws

Noether’s Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler‐Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. For the examples, we demonstrate, knowledge of this structure allows the Euler‐Lagrange equations to be integrated with relative ease. Our methods take advantage of recent advances in the theory of moving frames by Fels and Olver, and in the symbolic invariant calculus by Hubert. The results here generalize those appearing in Kogan and Olver [ 1 ] and in Mansfield [ 2 ]. In particular, we show results for high‐dimensional problems and classify those for the three inequivalent SL(2) actions in the plane.     

Analysis of the Lorenz Equations for Large r

In the limit of large r, the Lorenz equations become “almost” conservative. In this limit, one can use the method of averaging (or some equivalent) to obtain a set of two autonomous differential equations for two slowly varying amplitude functions B and D. A stable fixed point of these equations represents the stable periodic solution which is observed at large r. There is an invariant line B = D on which the method breaks down and the averaged equations are no longer valid. In this paper we show how to extend the validity of the analysis by Poincaré mapping B and D across this line. This extended analysis provides (in principl ) a complete recipe for constructing approximate solutions, and shows how a strange invariant set can occur in connection with an essentially analytically constructed two-dimensional mapping.     

Differential invariants of generic hyperbolic Monge-Ampère equations

In this paper basic differential invariants of generic hyperbolic Monge-Ampère equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.      

The existence of periodic solutions for nonlinear beam equations on by a para‐differential method

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the d‐dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para‐differential conjugation. Given the nonresonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.     

Young Tableaux and the Frobenius–Pullback of Tensor Differential Forms

In the case of prime characteristic char k = p > the pullback of the Frobenius provides an opportunity to define new discrete birational invariants of algebraic manifolds using the p^s–th powers (df) instead of the differentials df. Earlier results concerning T–symmetrical tensor forms on complete intersections will be generalized with regard to these differential forms.     

The Knizhnik-Zamolodchikov equations associated with the root system <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

At the end of the twentieth century, in mathematical physics, the Knizhnik-Zamolodchikov equations for the root systems A _n and their generalizations for the root systems of types B _n , C _n , and D were constructed. For the root system of type G ₂, the vector version of the Knizhnik-Zamolodchikov equations was obtained by M. P. Zamakhovskii and V. P. Leksin. However, the tensor version of these equations has remained unstudied. In this paper, the Knizhnik-Zamolodchikov equations associated with the root system of type G ₂ are considered. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Nonlinear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families

Given a group, its coset spaces provide all homogeneous spaces for its action. A subgroup chain allows for the construction of a bundle of sections over a coset space of independent variables, where the fiber coordinates are dependent variables and all their partial derivatives up to some order, (i.e., the kth order jet). In this coset bundle, group invariants take the form of differential equations. We present two families of group-subgroup chains, one leading to various tensor Burgers-type differential equations, and the other to Korteweg-de Vries equations with an nth space derivative. Maps of the Hopf-Cole type appear in both families as transformations which intertwine the original group action to a multiplier realization of a normally extended group, yielding a new differential equation with greater symmetry.     

Stepanov‐like doubly weighted pseudo almost automorphic processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion

This paper investigates the properties of the p‐mean Stepanov‐like doubly weighted pseudo almost automorphic (S^pDWPAA) processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion. We firstly prove the equivalent relation between the S^pDWPAA and Stepanov‐like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for S^pDWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the S^p DWPAA solution for a class of nonlinear Sobolev‐type stochastic differential equations driven by G‐Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.     

A Contact Linearization Problem for Monge-Ampère Equations and Laplace Invariants

We solve a problem of contact linearization for non-degenerate regular Monge-Ampère equations. In order to solve the problem we construct tensor invariants of equations with respect to contact transformations and generalize the classical Laplace invariants.      

Fast evaluation of nonlinear functionals of tensor product wavelet expansions

For a nonlinear functional f, and a function u from the span of a set of tensor product interpolets, it is shown how to compute the interpolant of f (u) from the span of this set of tensor product interpolets in linear complexity, assuming that the index set has a certain multiple tree structure. Applications are found in the field of (adaptive) tensor product solution methods for semilinear operator equations by collocation methods, or after transformations between the interpolet and (bi-) orthogonal wavelet bases, by Galerkin methods.