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The class of second-order ordinary differential equations y'' = A(x, y)y' + B(x, y) is studied by methods of the geometry of jet spaces and the geometric theory of differential equations. The symmetry group of this class of equations is calculated, and the field of differential invariants of its action on equations is described. These results are used to state and prove a criterion for the local equivalence of two nondegenerate ordinary differential equations of the form y'' = A(x, y)y' + B(x, y), inwhich the coefficients A and B are rational in x and y. [ABSTRACT FROM AUTHOR]

The article presents a mathematical analysis on general problems of contact and point classifications for functions defined on the n-jet space Jn&#211D; for an arbitrary n≿2. It describes the actions of contact and point pseudogroups on the n-jet space in the Lie alegra of the contact pseudogroup contact. It uses the methods of the theory of differential invariants in solving the problem.

We define two families of homogeneous ideals of the algebra of polynomials generated by power entries of the general matrix and its operator invariants. We study the combinatorial characteristics of these ideals and, in greater detail, the case of second order. [ABSTRACT FROM AUTHOR]

We construct asymptotic solutions of the Navier-Stokes equations describing the two-phase Taylor-scale structures consisting of periodic systems of localized vortex filaments. Such solutions are related to certain topological invariants of divergence-free vector fields on the two-dimensional cylinder or the torus. The equations describing the evolution of the vortex system are defined on a graph which is the set of trajectories of a divergence-free field. [ABSTRACT FROM AUTHOR]

The article explores how any normal subgroup with nonisotropic orbit in a transitive group of confomal transformations is inessential. Researchers have dispensed with the nonisotropy assumption at the expense of imposing a constraint on the dimension of the orbit. It is proven that any conformal transformation group either is essential or consists of homotheties with respect to the given metric on the manifold.

We investigate additive transformations on the space of real or complex matrices that are monotone with respect to any admissible partial order relation. A complete characterization of these transformations is obtained. In the real case, we show that such transformations are linear and that all nonzero monotone transformations are bijective. As a corollary, we characterize all additive transformations that are monotone with respect to certain classical matrix order relations, in particular, with respect to the Drazin order, left and right *-orders, and the diamond order. [ABSTRACT FROM AUTHOR]

In this paper, the norm of the Cauchy transform C is obtained on the space L2( D, dμ), where dμ = ω(| z|) dA( z). Also, (for the case ω ≡ 1), the first regularized trace of the operator C* C on L2(Ω) is obtained. The results are illustrated by examples, with different specific choices of the function ω and the domain Ω. [ABSTRACT FROM AUTHOR]

For the Born-Infeld equation in the hyperbolic domain of its solutions, we obtain first-order conservation laws depending on two arbitrary functions. It is shown that each conservation law is related to some Bäcklund transformation that transforms the Born-Infeld equation into some related equation. [ABSTRACT FROM AUTHOR]

The article focuses on the properties of invertible transformations, T : M → M preserving a finite measure μ on M, known as automorphisms of the measure space(M, Σ, μ). It assigns meanings to the generic and nongeneric properties of the transformations' asymptotic or algebraic properties. It also describes construction of transformations with "unusual" properties in ergodic theory using Rudolph's counterexample machine.