## Selected Scientific Discoveries upto now

### funded by ARO

Elementary proofs of Diaconis e Shashahani bounds on mixing times for the Bernoulli-Laplace diffusion model, and first passage times to stationarity. Our new proofs are based on explicit solutions of the eigenvectors of the transition matrix which is not self-adjoint, and extends the Pickering-Lim generating function method (see below refs) from linear transforms based on a subgroup of SL(2,R) to nonlinear transformations. We avoid any use of group representations of factor groups of S_n and the machinery of spherical functions and Gelfand pairs.

C. Lim and W. Pickering "Bounds for mixing times of the Bernoulli-Laplace model- Elementary proofs and calculations of first passage times to mean stationarity", submitted May 2017

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: For a sparse network such as over an ensemble of random graphs with fixed degree sequence, we showed that consensus times for voter model like games can be expressed as a product of two terms, namely, the complete graph consensus times which can be obtained explicitly by our generating function method in this paper and a term expressing the topology of the random graphs as a product or quotient resp. of the first two degree moments of the graphs, for the speaker first and listener first updates resp. A socially significant consequence is that the consensus times are longer for speaker first update than for listener first ones, thus providing some evidence for the adage "it is better to listen than to speak ...".

Pickering and C. Lim, "Solutions of Urn Models of Pairwise Interactions and Applications to Social, Physical and Biological Sciences", accepted by Phy. Rev E 2017

Applications of the Multi-zealots case and an Entropic measure of dividedness to the 2016 US Presidential Elections: 2 data points at least can be extracted from the current electoral process for crude comparisons to the relevant predictions of the naming games model in Network science. The consequences of the naming game theory are applicable to the 2016 presidential elections in the United States. The results of polling have shown that outsider candidates with unprecedented viewpoints have gained considerable popularity. For example, in January 2015, polls indicated that about 4% of Democratic voters supported Bernie Sanders, a self-proclaimed anti-establishment candidate. This figure has exhibited a steady increase over the course of about a year to 39% as of February of 2016. Outsider Republican candidate Donald Trump has exhibited a similar rise in popularity, starting from 4% of party supporters in April 2015 to 36% in February 2016. According to the Naming Game theory, this division of the majority opinions allows for even initially small committed groups to thrive. The theory of the Naming Game with zealots is a paradigm in the science of social revolution in the real world. It has been noted above that these principles can be applied in terms of the recent dynamic of the 2016 presidential election, with the anti-establishment candidates receiving unprecedented support from constituents. This support is made possible largely by the discontent and division from within the political parties themselves, and is a symptom of an overall social and ideological identity crisis. Although there are only two major political parties in the United States, each political party has created much disagreement among themselves, effectively dividing them into different factions that are united in name only. Political and social changes in the environment will have consequences on the initial condition in the context of the model, which are captured by the concept of an entropy of the initial population fractions supporting each candidate: January 2015 entropies are, for the Republicans 2.8, and for the Democrats much lower 1.7. From this initial data, we expect that the Democratic primary will coalesce somewhat earlier on a front runner than the Republican side. See W. Pickering, B. Szymanski, C. Lim, "Analysis of the high dimensional naming game with committed minorities", arXiv preprint arXiv:1512.03390, 2015 click here to downloadPapers and Preprints on 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 neutrals in social forums - exact soln of 3 state voter model", arXiv preprint arXiv:1411.0530, 2014 click here to download

Published as "Solution of the multistate voter model and application to strong neutrals in the naming game", 1 March 2016 issue of Physical Review E (Vol.93, No.3): DOI: 10.1103/PhysRevE.93.032318 click here to download[4b] W. Pickering, B. Szymanski, C. Lim, "Analysis of the high dimensional naming game with committed minorities", arXiv preprint arXiv:1512.03390, 2015 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. Published as," Social opinion dynamics is not chaotic", Int. J. Mod. Phys. B DOI: 10.1142/S0217979215410064

[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.

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).

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.