Simulation of harmonic oscillators on the lattice

This work deals with the simulation of a two-dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second-order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state-of-the-art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two-dimensional lattice and check its energy conservation.     

Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let \((M,g_0)\) be a smooth compact manifold of dimension \(n\ge 3\) with boundary. Given any smooth functions f in M and h on \(\partial M\), does there exist a conformal metric of \(g_0\) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant \(Q(M,\partial M)\) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

Convex underestimators of polynomials

Convex underestimators of a polynomial on a box. Given a non convex polynomial ${f\in \mathbb{R}[{\rm x}]}$ and a box ${{\rm B}\subset \mathbb{R}^n}$ , we construct a sequence of convex polynomials ${(f_{dk})\subset \mathbb{R}[{\rm x}]}$ , which converges in a strong sense to the “best” (convex and degree-d) polynomial underestimator ${f^{*}_{d}}$ of f. Indeed, ${f^{*}_{d}}$ minimizes the L ₁-norm ${\Vert f-g\Vert_1}$ on B, over all convex degree-d polynomial underestimators g of f. On a sample of problems with non convex f, we then compare the lower bounds obtained by minimizing the convex underestimator of f computed as above and computed via the popular α BB method and some of its other refinements. In most of all examples we obtain significantly better results even with the smallest value of k.     

Deficiency zero for random reaction networks under a stochastic block model framework

Journal of Mathematical Chemistry - Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper...     

BowtieArene: A Dual Macrocycle Exhibiting Stimuli-Responsive Fluorescence

Although highly useful in supramolecular chemistry, pillararenes lack a fluorophore in their skeleton. Here we present BowtieArene, a novel fluorescent dual macrocycle, featuring a central tetraphenylethylene-derived fluorophore and two pillar-like, pentagon-shaped cavities which are comparable to pillar[5]arene. This concisely prepared, figure-of-eight molecule exhibits vapor absorption and host–guest capabilities, as well as intriguing switchable fluorescence. The fluorochromism of BowtieArene can be triggered by multiple external stimuli including solvent, vapor, and mechanical force, with excellent reversibility and stability. Experimental and theoretical evidence indicate that the fluorochromism should be closely related to molecular packing.