Graphene-Enhanced Optical Signal Processing

Graphene has emerged as an attractive material for a myriad of optoelectronic applications due to its variety of remarkable optical, electronic, thermal and mechanical properties. So far, the main focus has been on graphene based photonics and optoelectronics devices. Due to the linear band structure allowing interband optical transitions at all photon energies, graphene has remarkably large third-order optical susceptibility χ(3), which is only weakly dependent on the wavelength in the near-infrared frequency range. Graphene possesses the properties of the enhancement four-wave mixing (FWM) of conversion efficiency. So, we believe that the potential applications of graphene also lies in nonlinear optical signal processing, where the combination of its unique large χ(3) nonlinearities and dispersionless over the wavelength can be fully exploited. In this chapter, we give a brief overview of our recent progress in graphene-assisted nonlinear optical device which is graphene-coated optical fiber and graphene-silicon microring resonator and their applications, including degenerate FWM based tunable wavelength conversion of quadrature phase-shift keying (QPSK) signal, two-input optical computing, three-input high-base optical computing, graphene-silicon microring resonator enhanced nonlinear optical device for on-chip optical signal processing, and nonlinearity enhanced graphene-silicon microring for selective conversion of flexible grid multi-channel multi-level signal

Numerical characterization of the hard Lefschetz classes of dimension two

We study the numerical characterization of two dimensional hard Lefschetz classes given by the complete intersections of nef classes. In Shenfeld and van Handel's breakthrough work on the characterization of the extremals of the Alexandrov-Fenchel inequality for convex polytopes, they proposed an open question on the algebraic analogue of the characterization. By taking further inspiration from our previous work with Shang on hard Lefschetz theorems for free line bundles, we formulate and refine the conjectural picture more precisely and settle the open question when the collection of nef classes is given by a rearrangement of supercriticality, which in particular includes the big nef collection as a special case. The main results enable us to refine some previous results and study the extremals of Hodge index inequality, and more importantly provide the first series of examples of hard Lefschetz classes of dimension two both in algebraic geometry and analytic geometry, in which one can allow nontrivial augmented base locus and thus drop the semi-ampleness or semi-positivity assumption. As a key ingredient of the numerical characterization, we establish a local Hodge index inequality for Lorentzian polynomials, which is the algebraic analogue of the local Alexandrov-Fenchel inequality obtained by Shenfeld-van Handel for convex polytopes. This result holds in broad contexts, e.g., it holds on a smooth projective variety, on a compact K\"ahler manifold and on a Lorentzian fan, which contains the Bergman fan of a matroid or polymatroid as a typical example.Comment: 37 pages; comments welcome! arXiv admin note: text overlap with arXiv:2011.04059 by other author

Intersection theoretic inequalities via Lorentzian polynomials

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to $m$-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for $m$-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by $m$-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Pl\"{u}nnecke-Ruzsa inequalities) from additive combinatorics in our contexts.Comment: 27 pages; comments welcome

Hard Lefschetz properties, complete intersections and numerical dimensions

We study the positivity of complete intersections of nef classes. We first give a sufficient and necessary characterization on the complete intersection classes which have hard Lefschetz property on a compact complex torus, equivalently, in the linear case. In turn, this provides us new kinds of cohomology classes which have Hodge-Riemann property or hard Lefschetz property on an arbitrary compact K\"ahler manifold. We also give a complete characterization on when the complete intersection classes are non-vanishing on an arbitrary compact K\"ahler manifold. Both characterizations are given by the numerical dimensions of various partial summations of the given nef classes. As an interesting byproduct, we show that the numerical dimension endows any finite set of nef classes with a loopless polymatroid structure.Comment: comments welcome

Epidemic modelling by ripple-spreading network and genetic algorithm

Mathematical analysis and modelling is central to infectious disease epidemiology. This paper, inspired by the natural ripple-spreading phenomenon, proposes a novel ripple-spreading network model for the study of infectious disease transmission. The new epidemic model naturally has good potential for capturing many spatial and temporal features observed in the outbreak of plagues. In particular, using a stochastic ripple-spreading process simulates the effect of random contacts and movements of individuals on the probability of infection well, which is usually a challenging issue in epidemic modeling. Some ripple-spreading related parameters such as threshold and amplifying factor of nodes are ideal to describe the importance of individuals’ physical fitness and immunity. The new model is rich in parameters to incorporate many real factors such as public health service and policies, and it is highly flexible to modifications. A genetic algorithm is used to tune the parameters of the model by referring to historic data of an epidemic. The well-tuned model can then be used for analyzing and forecasting purposes. The effectiveness of the proposed method is illustrated by simulation results

Hard Lefschetz theorems for free line bundles

We introduce a partial positivity notion for algebraic maps via the defect of semismallness. This positivity notion is modeled on $m$-positivity in the analytic setting and $m$-ampleness in the geometric setting. Using this positivity condition for algebraic maps, we establish K\"ahler packages, that is, Hard Lefschetz theorems and Hodge-Riemann bilinear relations, for the complete intersections of Chern classes of free line bundles.Comment: 14 pages; comments welcome