## Selected Scientific Discoveries upto 2013

### funded by ARO

Generalized Jacobi Coordinates based on a graph-theoretic algorithm on subclass of sign-nonsingular patterns involving Binary Trees (also related to the Kastelyn-Percus theorems for counting perfect matchings and Exact combinatorial solutions of the 2D Ising Model) and used to simplify the calculations of the KAM-nondegeneracy condition in N-body problems, have recently appeared in two different directions - (1) Entangled Fock Space in Einstein-Podolsky-Rosen states (J. Katriel et al Molecular Phys 2008), (2) Shape Dynamics in classical and quantum gravity (cf J. Barbour et al Perimeter Institute) :

C. Lim, Binary trees, symplectic matrices and the Jacobi coordinates of celestial mechanics, Arch. Rat. Mech. Anal. 115 (1991), 153-165; C. Lim, Canonical transformations and graph theory, Physics Letters A, 138 1989, 258-266.

Exact solutions of an energy-enstrophy theory (based on Kac's spherical model) in the statistical physics of quasi-2D flows, using the newly discovered equivalence of the canonical and microcanonical ensembles for long-range interactions on compact oriented manifolds (via Hodge Theory), leads to results that are applicable to Planetary Atmospheres in extra-solar systems such as "Super-rotators are more common than Sub-rotators in Nearly Barotropic Planetary Atmospheres", and to the enigmatic super-rotation of Venus and Titan:

C. C. Lim, "Phase Transition to Super-rotating Atmospheres in a Simple Planetary Model for a Non-Rotating Massive Planet - Exact Solution", Physical Review E 86 (6)(2012) click here to download from APS.

For details on the rotating case, see submitted, 2013.Anomalous Expansion and Negative Specific Heat in quasi-2D trapped vortex filament bundles in an unbounded horizontal domain was discovered by Tim Andersen and Chjan Lim and applied to the confinement and stability of hot Tokamak plasmas; see for example:

T. Andersen and C. Lim, "Negative Specific Heat in a Quasi-2D Generalized Vorticity Model'', Phys. Rev. Lett. 99, 165001, Oct 2007.

Tipping Points in Social Opinion Dynamics PRE 11:

J. Xie, S. Sreenivasan, G. Korniss, W. Zhang, C. Lim, and B.K. Szymanski, "Social consensus through the influence of committed minorities", Physical Review E 84, 011130 (2011).

Time-Reversible Dynamical System, see for example:

M. Krupa, C. Lim, and M. Golubitsky, "Time-reversibility and particle sedimentation'', SIAM J. Applied Math. 51(1), 49-72 (1991).

Other resuts 0n Network Science and Applied Probability:

[1] W. Zhang and C.C. Lim, "The Concentration and Stability of the Community Detecting Functions on Random Networks", Internet Math, Accepted December 2012, appeared online March 2013. click here to download

[2] W. Zhang, C.C. Lim and B. Szymanski, "Analytic Treatment of Tipping Points for Social Consensus in Large Random Networks", Phys. Rev. E, Appeared December 2012, Phys. Rev. E. December 2012

[3] W. Zhang, Korniss, Szymanki, C.C. Lim, "Spatial Propagation of Opinions: Naming games on random geographic graphs", to appear online Sci Reports 2014, Sci Reports

[4] C.C. Lim and W. Zhang, "Monotonicity of Social Opinion Dynamics on Large Networks", submitted July 31 2013, for the role of monotonicity in math. sociology where pure diffusion is secondary to drift.

[5] A. Thompson, B. Szymanski, C.C. Lim, "Propensity and Stickiness in the Naming Games", submitted, on robustness of tipping points in two para. family of social opinion models.