Mean field equations, hyperelliptic curves and modular forms: II

A pre-modular form $Z_n(\sigma; \tau)$ of weight $\tfrac{1}{2} n(n + 1)$ is introduced for each $n \in \Bbb N$, where $(\sigma, \tau) \in \Bbb C \times \Bbb H$, such that for $E_\tau = \Bbb C/(\Bbb Z + \Bbb Z \tau)$, every non-trivial zero of $Z_n(\sigma; \tau)$, namely $\sigma \not\in E_\tau[2]$, corresponds to a (scaling family of) solution to the mean field equation \begin{equation} \tag{MFE} \triangle u + e^u = \rho \, \delta_0 \end{equation} on the flat torus $E_\tau$ with singular strength $\rho = 8\pi n$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(\tau) \subset {\rm Sym}^n E_\tau$, the Lam\'e curve, associated to the MFE was constructed. Our construction of $Z_n(\sigma; \tau)$ relies on a detailed study on the correspondence $\Bbb P^1 \leftarrow \bar X_n(\tau) \to E_\tau$ induced from the hyperelliptic projection and the addition map. As an application of the explicit form of the weight 10 pre-modular form $Z_4(\sigma; \tau)$, a counting formula for Lam\'e equations of degree $n = 4$ with finite monodromy is given in the appendix (by Y.-C. Chou).Comment: 32 pages. Part of content in previous versions is removed and published separately. One author is remove

Analytic Aspects of the Toda System: II. Bubbling behavior and existence of solutions

In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for $SU(N+1)$. We give a much more precise bubbling behavior of solutions and study its existence in some critical casesComment: 33 pages, to appear in Comm. Pure Appl. Mat

A coordinated approach for supply-chain tracking in the liquefied natural gas industry

With the increased size and complexity of liquefied natural gas (LNG) projects, supplychain management has become a challenging process due to involvements of the remote location of the project site and the multiple stakeholders. The transparency and traceability of the supply-chain are critical as any surpluses or shortages of materials will put the project at risk. Currently, limited research has been conducted on LNG projects considering the total supply-chain perspective, which refers to all stages of materials tracking in off-site manufacturing, transportation, and site logistics. The purpose of this research is to propose a framework of a coordinated approach for supply-chain tracking in the LNG industry. Two focus group studies were organized to develop the proposed framework: One for LNG construction supply chain process development, and another for alternative tracking technologies selection. In addition, two experiments, namely off-site fabrication tracking and site logistics tracking, were conducted in a field to evaluate the feasibility of the proposed framework. Technology limitations were also discussed in terms of field implementation