arXiv research

Proves a theorem for normal distributions on manifolds with boundary.

Theorem proves integrability for piecewise-smooth distributions.

Study h-principles for non-integrable distributions on manifolds.

Integrates rough geometric forms on manifolds.

In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connect…

We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provi…

The paper uses Fourier integral theorem for estimating multivariate distributions.

We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…

The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.

The paper proves integral formulas for manifolds with multiple orthogonal distributions.

This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. T…

Continuous-time distributed mirror descent with integral feedback converges to global optimum.

We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector fields. We develop and apply an algorithm to find the largest integrable sub-distr…

A theorem proves integrability of Fréchet tangent distributions.

We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability tensors of the distributions. The formulas under some conditions yield splitting…

Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.

For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $Δ$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2,…

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is pro…

This paper solves a complex differential relation using a novel 'avoidance trick'.

Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.

We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive…

Paper studies invariant distributions of bi-Hamiltonian structures.

By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i…

Paper proposes efficient method to calculate Fisher-Bingham distribution normalizing constant.

Paper converts quantiles to cumulative distribution functions to simplify risk measures.

Develops methods for integrating multivariate normals and computing classification measures.

RHMC accelerates sampling from log-concave distributions.

New integral defined for Hölder continuous functions, characterizing distributional volume forms.

Integrable geodesics found on special orthogonal group.

In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijectiv…

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical theorem of Frobenius Theorem which says that an involutive C^1 distribution is uniquely integrable.

Develops integrators for nonholonomic systems on Lie groups.

New definition of Bäcklund transformation for surface isometric deformation.

Study on integrability of geodesic flows on Heisenberg group.

Study finds homogeneous spaces with geodesic orbits but no integrable distributions.

A new method calculates fractional moments using the moment-generating function.

We give a conceptual proof of the fact that if M is a complete submanifold of a space form, then the maximal integral manifolds of the nullity distribution of its second fundamental form through points of minimal index of nullity are complete.

Simplified uHMC with time integration improves accuracy and efficiency.

Integrable LCK manifolds characterized as Kähler Lie algebras.

We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on projective spaces.

We give necessary and sufficient conditions for the real distributions defined by a metallic pseudo-Riemannian structure to be integrable and geodesically invariant, in terms of associated tensor fields to the metallic structures and of adapted connections. In the integrable case, we prove a Chen-type inequality for th…

We study positive definite quaternionic contact $(4n+3)$-manifolds ($qc$-manifold for short). Just like the $CR$-structure contains the class of Sasaki manifolds, the $qc$-structure admits a class of $3$-Sasaki manifolds with integrable distribution isomorphic to $\mathfrak{su}(2)$. A big difference concerning the inte…

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…

We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…

The paper establishes a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.

Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…

A new method for reconstructing flows from perturbed distributions.

Study Hochschild cohomology of dg manifolds linked to integrable distributions.