Dimensional reduction to hypersurface of foliation

On the Characteristic Foliation on a Smooth Hypersurface in a Holomorphic Symplectic Fourfold

This terminology is explained by Bogomolov decomposition theorem which states that, up to a finite étale covering, each holomorphic symplectic manifold is a product of several irreducible ones and a torus. Let X be a holomorphic symplectic manifold with a holomorphic symplectic form ω. Let D be a smooth divisor on X. At each point of D, the restriction of ω to D has one-dimensional kernel. This...

HyperSurface Classifiers Ensemble for High Dimensional Data Sets

Based on Jordan Curve Theorem, a universal classification method called HyperSurface Classifier (HSC) has recently been proposed. Experimental results show that in three-dimensional space, this method works fairly well in both accuracy and efficiency even for large size data up to 10. However, what we really need is an algorithm that can deal with data not only of massive size but also of high ...

Smooth Maps Transverse to a Foliation

Dimensional reduction and quantum-to-classical reduction at high temperatures.

We discuss the relation between dimensional reduction in quantum field theories at finite temperature and a familiar quantummechanical phenomenon that quantum effects become negligible at high temperatures. Fermi and Bose fields are compared in this respect. We show that decoupling of fermions from the dimensionally reduced theory can be related to the non-existence of classical statistics for ...

Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction

A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional gr...