## Selected Scientific Discoveries upto now

### funded by ARO

Piano pieces for the Holidays May 2010 Debussy and Beethoven

Quasi MC methods have applications in quantitative finance and fintech. Fast random numbers generation and robust algorithms for generating highly uniform sequences are essential to these applications and to emerging ones in quantum information science. This article offers a new physical process and algorithm for generating sequences with circular discrepancy (a measure of their uniformness property) that are comparable to wellknown ones such as Sobol.

S.M. Assad and C. Lim "Circular discrepancy and a MC Algo for gennerating low circular discrepancy sequences", chap 1 in Krause, et al eds, Vortex Dominated Flows: Dedicated to Lu Ting's 80th birthday, pages 1-15,World Sci Pubs, 2005.

A new method for the organic growth of networks based on the concepts of Relevance and Importance extends the original methof of Preferential Attachment (PA) due to Barabasi et al. We calculate the network statistics of such RIPA - networks and provide examples from the growth of cities and travel networks of our models.

W. Zhang and C. Lim "Network Evolution by RIPA", November 2018

A new nonlinear transform to diagonalize any finite size Bernoulli-Laplace model extends the method of triangularization of certain Markov chain models proposed by Lim and Pickering recently. All eigenvectors of the BL problem are calculated in closed form in this easy to use method. Compared to other known methods for the BL model, including the group representation method of Diaconis and Shashahani and the orthogonal polynomials method of Karlin and McGregor, our method reveals directly from symmetries, the correct transform of independent variables needed to reduce the linear partial differential operator associated with the BL transition matrix to explicit form. This LPDO acts on a homogeneous polynomial in monomials of indeterminates or independent variables whose coefficients encode all the right and left eigenvectors of the BL matrix.

C. Lim and W. Pickering "A nonlinear transform for the diagonalization of the Bernoulli-Laplace diffusion model and orthogonal polynomials", November 2018

Symplectic transformations to relative Jacobi coords generated by full binary trees and its recent applications in the complex manifold theory of Choreographic and periodic solutions in Vortex N-body dynamics.

See Qun Wang's recent article in Archive for Rat Mech and Analysis and his preprint "Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of -Vortex Type Q Wang" - arXiv preprint arXiv:1811.06595, 2018 - arxiv.org.

See also C. Lim's "Binary trees, symplectic matrices and the Jacobi coordinates of celestial mechanics", Arch. Rat. Mech. Anal. 115 (1991), 153-165; "Graph theory and a special class of symplectic transformations: the generalized Jacobi variables", J. Math. Phy 32(1), 1-7, (1991); "Canonical Transformations and Graph Theory", Phys Lett A 138 1989, 258-266.

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

W. Pickering, B. Szymanski, C. Lim, "Analysis of the high dimensional naming game with committed minorities", arXiv preprint arXiv:1512.03390, 2015

Papers 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.