Alexandrov proved that any simplicial complex homeomorphic to a sphere with strictly non-negative Gaussian curvature at each vertex can be isometrically embedded uniquely in {{\mathbb {R}}}^{3} as a convex polyhedron. Due to the nonconstructive nature of his proof, there have yet to be any algorithms, that we know of, that realizes the Alexandrov embedding in polynomial time. Following his proof, we developed the adiabatic isometric mapping (AIM) algorithm. AIM uses a guided adiabatic pull-back procedure on a given polyhedral metric to produce an embedding that approximates the unique Alexandrov polyhedron. Tests of AIM applied to two different polyhedral metrics suggests that its run time is sub cubic with respect to the number of vertices. Although Alexandrov's theorem specifically addresses the embedding of convex polyhedral metrics, we tested AIM on a broader class of polyhedral metrics that included regions of

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We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases.

Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional N = 2 SCFT. The SeibergWitten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding SeibergWitten differential is given by the GelfandLeray form of K. Saitos primitive form. Our result also extends the SeibergWitten solution to include irrelevant deformations.