## Selected Scientific Discoveries upto 2015

### funded by ARO

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 of committed minority fraction in Social Opinion Dynamics mapped to the Saddle-Node and Pitchfork Bifurcations with sociological significance in the robust smallness of the tipping fraction of 10 percent; application of monotone dynamical systems to the social interactions of tribes:

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).

Extending Kac's 1947 generating function solution of the Ehrenfest Dog Flea model, Pickering and Lim solved exactly a large family of 2 Urns-2 Particles stochastic models by diagonalizing pentadiagonal markov transition matrix Pij; this family includes variants of the Voter model, Moran's genetic drift model and all discrete-time birth death processes solved by the Karlin-McGregor (1955-1962) correspondence between these processes (all of which have tridiagonal Pij) and the Stieltjes Moment Problem of Orthogonal systems of Polynomials. Our method works for stochastic Urn models that are not the standard random walks or birth-death processes with tridiagonal Pij - we can diagonalize a large class of pentadiagonal Pij:

Pickering and C. Lim, "Solutions of Urn Models", submitted 2015

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 results 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 Mathematics 9 (4), 360-383, 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 86 (6), 061134, 2012

[3] W. Zhang, Korniss, Szymanki, C.C. Lim, "Spatial Propagation of Opinions: Naming games on random geographic graphs", Sci Reports 2014, Scientific Reports 4, 5568 (2014) and doi: 10.1038/srep05568 Sci Reports

[4] W. Pickering, C.C. Lim, "Solution of Voter models by Spectral Analysis", click here to download

Physical Review E (Vol.91, No.1): DOI: 10.1103/PhysRevE.91.012812[4a] C.C. Lim, W. Pickering, "Information sharing of strong neitrals in social forums - exact soln of 3 state voter model", arXiv preprint arXiv:1411.0530, 2014 click here to download

[5] 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.

[6] A. Thompson, B. Szymanski, C.C. Lim, "Propensity and Stickiness in the Naming Games", Phys Rev E 90, 042809, 2014 on robustness of tipping points in two para. family of social opinion models.