This work concerns the problem of relating characteristic numbers of onedimensional holomorphic foliations of PC to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a one-dimensional holomorphic foliation F of M is a morphism Φ : L −→ TM where L is a holomorphic line bundle on M . The singular set of F is the analytic subvariety sing(F) = {p ...

It is shown that the classical laws of thermodynamics require that mechanical systems must exhibit energy that becomes unavailable to do useful work. In thermodynamics, this type of energy is called entropy. It is further shown that these laws require two metrical manifolds, equations of motion, field equations, and Weyl’s quantum principles. Weyl’s quantum principle requires quantization of th...

This paper is concerned with fixed-point free S-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which...

A solution manifold is the collection of points in a d-dimensional space satisfying system s equations with s<d. Solution manifolds occur several statistical problems including missing data, algorithmic fairness, hypothesis testing, partial identifications, and nonparametric set estimation. We theoretically algorithmically analyze manifolds. In terms theory, we derive five useful results: smoot...

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by m...

‎The object of the present paper is to introduce and study a type of non-flat semi-Riemannian manifolds‎, ‎called‎, ‎super generalized recurrent manifolds which generalizes both the notion of hyper generalized recurrent manifolds [‎A.A‎. ‎Shaikh and A‎. ‎Patra‎, On a generalized class of recurrent manifolds‎, Arch‎. ‎Math‎. ‎(Brno) 46 (2010) 71--78‎.] and weakly generalized recurrent manifolds ...

Information geometry studies the dually flat structure of a manifold, highlighted by the generalized Pythagorean theorem. The present paper studies a class of Bregman divergences called the (ρ, τ)-divergence. A (ρ, τ)-divergence generates a dually flat structure in the manifold of positive measures, as well as in the manifold of positive-definite matrices. The class is composed of decomposable ...

Abstract We explore the approximation of attracting manifolds of complex systems using dimension reducing methods. Complex systems having high dimensional dynamics typically are initially analyzed by exploring techniques to reduce the dimension. Linear techniques, such as Galerkin projection methods, and nonlinear techniques, such as center manifold reduction are just some of the examples used ...