In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic γ with a single double point is asymptotic to Ldim(Teichmuller space of S.). Since closed geodesics with one double point fall into a finite number of M≀⌈(S) orbits, we get the same asympotic estimate for the number of such geod...

In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesics on the manifold. Here we do not discuss methods to compute these geo...

In this work we consider a given root of family n-degree polynomials as one-variable function that depends only on the independent term. Then prove satisfies several ordinary differential equations (ODE). More concretely, it simple separated variables ODE, first order generalized Abel ODE degree n−1 and an (n−1)-th linear ODE. Although some our results are not new, approach is self-contained. F...

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