We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to R^3. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together,these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-timesymmetric version of a theorem ofMeeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to R^3 contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to R^3 minus a finite number of open balls. An extension to higher dimensions is also discussed.

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A fundamental solution of the acoustical equation with a variable refraction coefficient is constructed. The solution satisfies the limiting absorption and radiation conditions. The optimal high frequency estimate is proved for square means of the solution. The source function for the diffusion equation is a by-product of this construction.

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If$G$is a 2-connected graph of order$n$and max{$d$($u$),$d$($v$)}≥$n$/2 for each pair of vertices$u$and$v$with distance$d$($u$,$v$)=2, then$G$is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if$G$is a 2-connected graph and |$N$($u$)∪$N$($v$)|+δ($G$)≥$n$for each pair of nonadjacent vertices$u$,$v$∈$V$($G$), then$G$is Hamiltonian. This paper generalizes the above results when$G$is 3-connected. We show that if$G$is a 3-connected graph of order$n$and max{|$N$($x$)∪$N$($y$)|+$d$($u$),|$N$($w$)∪$N$($z$)|+$d$($v$)}≥$n$for every choice of vertices$x$,$y$,$u$,$w$,$z$,$v$such that$d$($x$,$y$)=$d$($y$,$u$)=$d$($w$,$z$)=$d$($z$,$v$)=$d$($u$,$v$)=2 and where$x$,$y$and$u$are three distinct vertices and$w$,$z$and$v$are also three distinct vertices (and possibly |{$x$,$y$}∩{$w$,$z$}| is 1 or 2), then$G$is Hamiltonian.