This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints hα with arbitrary coefficients. The main purpose of the present paper is to make clear that classical dynamics of a totally constrained system is nothing but the fo...

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be an oriented immersed minimal Lagrangian submanifold of N without boundary and V be a holomorphic vector field in a neighbourhood of L in N . Let div(V ) be the (complex) divergence of V . Then the integral ∫ L div(V ) = 0. Vice versa suppose that N is Kahler-Einstein with non-zero scalar curvature a...

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassman bundle of p-planes Gr(p, R) → R by p-dimensional submanifolds that are integrals of the canonical contact differential system. The equivalence method yields an sl(n + 1, R)valued Cartan connection whose curvature captures the geometry of such foliation. In the flat case, the space of leaves o...

Twenty years ago it was not obvious that subspace approximations would be such a successful representation for faces and other phenomena whose measurement-space manifolds exhibit clear nonlinearities. Now PCA is ubiquitous in computer vision and, although its globally linear view of the data manifold can be a liability as systems scale up, it is not obvious that one can reliably construct a bet...

We consider a sequence of quantum states, $\rho_N$, associated with submanifold $\Lambda$ product two integral compact Kahler manifolds. show that, when is submanifold, then, in the semiclassical limit, these states are not entangled. discuss whether geometry (specifically being submanifold) has any relationship entanglement properties $\rho_N$.

Recent years have witnessed a resurgence of twistor methods in both mathematics and physics. On the physics side, this development has primarily been driven by Witten’s discovery [21] that strings in twistor space can be used to calculate Yang-Mills scattering amplitudes, resulting in concrete experimental predictions that can be tested using existing particle accelerators. Here, one of the mai...

Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much ...

Let N be a compact, orientable hyperbolic 3-manifold with ∂N a connected totally geodesic surface of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy: either N has a long return path (defined by Kojima-Miyamoto), or N has an embedded codimension-0 submanifold X with incompressible boundary T t ∂N , where T i...