arXiv research

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

Paper extends Poincaré's work to stochastic differential equations.

Study characterizes Finsler metrics with first integrals using specific curvature tensors.

New vector fields integrate first-order ODEs.

Method finds differential equations for integrable billiard tables.

The paper studies algebraic relations of first integrals on specific Lie groups.

Study finds first integrals in Finsler metrics with vanishing χ-curvature.

Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.

The paper defines and proves the existence of decompositions of integral varifolds.

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemanni…

Survey uses Milnor fibrations to classify first integrals of differential systems.

First-order ODEs linked to flat surfaces, leading to integrability.

New theorem links symmetries to first integrals in plasma physics.

We give various estimates of the first eigenvalue of the $p$-Laplace operator on closed Riemannian manifold with integral curvature conditions.

The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations i…

We study $n$-dimensional Kähler manifolds whose geodesic flows possess $n$ first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a manifold an $n$-dimensional commutative Lie algebra of infinitesimal automorphisms. This,…

New tensors help solve magnetic flow integrability.

In this work, we show an injectivity result and support theorems for integral moments of a m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of m-tensor field were first introduced by Sharafutdinov. At first we generalize a Helgason type support theorem proven by Krishnan and Stefanov in …

Complete integrability proved for SR geodesic flow on S^7.

This article is the continuation of our first article (math/9901028). It shows how the zero-anomaly result of Yang implies the equality between the configuration space integral and the Kontsevich integral.

Paper proves flexibility of specific relations using convex integration.

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows…

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kähler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In…

In this note we find a 6-dimensional h-spaces of the $[(21...1)(21...1)...(1...1)]$ type and then determine quadratic first integrals of the geodesic equations of these h-spaces.

This paper proves integrability of Birkhoff billiards inside convex cones.

In this work we study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. Also examples of integrable geodesic flows and of non-integrable ones are shown.

Paper extends Noether's Theorem to nonholonomic systems, proving conserved momentum.

The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.

We prove a regularity result for unit volume conformal metrics with integral scalar curvature bounds for $p>n/2$ and first eigenvalue of $Δ$ bounded from below by a constant $B > Λ_1(S^n,[g_{st.}]).$

Relates geodesic integrals to Killing tensors, exploring their dimensions.

Study on equilibrium points of dynamical systems with multiple integrals.

The first author conjectured certain relations for Morita-Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann-Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.

The study finds bounds for the first eigenvalue of the p-Laplacian on submanifolds.

The paper finds new metrics for geodesic flows with rational integrals.

Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…

Multipeakons are special solutions to the Camassa-Holm equation described by an integrable geodesic flow on a Riemannian manifold. We present a bi-Hamiltonian formulation of the system explicitly and write down formulae for the associated first integrals. Then we exploit the first integrals and present a novel approach…

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

Killing tensor fields have been thought of as describing hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, and thus solving the de…

We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.

This paper presents a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in t…

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlle…

Quadratic Killing tensors on Lie groups are always decomposable.

In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywh…

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.

Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.

The geodesic flow of a Riemannian metric on a compact manifold $Q$ is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle $T^*Q\setminus{Q}$. If the geodesic flow is toric integrable, the cosphere bundle admit…

Study calculates first $p$-widths of unit disk.