Geodesics are the shortest curves between any two points on a given surface. in state space of quantum systems play an important role theory geometric phases, as these also along which acquired phase is zero. Null (NPCs) generalization geodesics, defined zero even though they need not be points. Here we present decomposition geodesics and NPCs higher-dimensional space, allows understanding intr...

Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest non-separating cycle in 2O(g)n log log n time. Similar algorithms are given to compute a shortest...

We consider brane cosmology studying the shortest null path on the brane for photons, and in the bulk for gravitons. We derive the differential equation for the shortest path in the bulk for a 1+4 cosmological metric. The time cost and the redshifts for photons and gravitons after traveling their respective path are compared. We consider some numerical solutions of the shortest path equation, a...

let ? be an open connected subset of the complex plane c and let t be a bounded linear operator on a hilbert space h. for ? in ? let e the orthogonal projection onto the null-space of t-?i . we discuss the necessary and sufficient conditions for the map ?? to b e continuous on ?. a generalized gram- schmidt process is also given.