The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Workshop on Geodesics in August 2010 at the Chern Institute of Mathematics in Tianjin.

Remission is a new research outcome indicating long-term wellness. Remission not only sets a standard for minimal severity of symptoms and signs (resolution); it also sets a standard for how long symptoms and signs need to remain at this minimal level (6 months). Individuals who achieve remission from schizophrenia have better subjective well-being and better functional outcomes than those who do not. Research suggests that remission can be achieved in 20–60% of people with schizophrenia. There is some evidence of the usefulness of remission as an outcome indicator for clinicians, service users and their carers. This article reviews the literature on remission in schizophrenia and asks whether it could be a useful clinical standard of well-being and a foundation for functional improvement and recovery.

We have developed a three-dimensional (3D) circuit integration technology that exploits the advantages of silicon-on-insulator (SOI) technology to enable wafer-level stacking and micrometer-scale electrical interconnection of fully fabricated circuit wafers. This paper describes the 3D technology and discusses some of the advanced focal plane arrays that have been built using it.

For any \varepsilon > 0, we construct an explicit smooth Riemannian metric on the sphere S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is \varepsilon-dense in the unit tangent bundle. Moreover, for any \varepsilon > 0, we construct a smooth Riemannian metric on S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than \varepsilon.

Let $M$ be a compact $C^{\infty}$ Riemannian manifold.Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number ofgeodesic segments joining $p$ and $q$ with length $\leq T$.Mañé showed in [7] that\[\lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq= h_{\rm top},\]where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of$M$.

In this paper we exhibit an open set of metrics on the two-sphere for which\[\limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top},\]for a positive measure set of $(p,q)\in M\times M$.This answers in the negative questions raised by Mañé in[7].

We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.

We obtain a family of metrics on the two-dimensional sphere whose geodesic flow is ergodic and Bernoulli. This family includes real analytic metrics.

It is shown that a C2 flow on a compact three-dimensional manifold that preserves a smooth measure and has a continuous family of cones satisfying a certain invariance property must be ergodic.

The Reagan administration’s arguments purporting to justify the invasion of Grenada under international law must not be allowed to inveigle the American people into supporting this violent intervention into the domestic affairs of another independent state. Throughout the 20th century, the U.S. Government has routinely concocted evanescent threats to the lives and property of U.S. nationals as pretexts to justify armed interventions into sister American states. The transparency of these pretexts was just as obvious then as it is now. The Reagan administration has not established by means of clear and convincing evidence that there did in fact exist an immediate threat to the safety of U.S. citizens in Grenada. Even then, such a threat could have justified only a limited military operation along the lines of the Israeli raid at Entebbe for the sole purpose of evacuating the major concentration of U.S. nationals at the medical college.

It is shown that the unit tangent bundle of a compact uniform visibility manifold with no focal points contains a subset of positive Liouville measure on which all the characteristic exponents of the geodesic flow (except in the flow direction) are non-zero. This completes Pesin's proof that the geodesic flow of such a manifold is Bernoulli.