Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_\alpha$ denotes Riesz potential and $\alpha \in (0, N)$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $\alpha \to 0$ or $\alpha \to N$. This leads to the uniqueness and nondegeneracy of ground states when $\alpha$ is sufficiently close to $0$ or close to $N$.Comment: 18 pages, revised versio

Micro-Assembly using Elastomeric Surfaces with Switchable Dry Adhesion

This work presents a micromanufacturing method for constructing microsystems, which we term ‘micro-masonry’ based on individual manipulation, influenced by strategies for deterministic materials assembly using advanced forms of transfer printing. Analogous to masonry in construction sites, micro-masonry consists of the preparation, manipulation, and binding of microscale units to assemble microcomponents and microsystems. We used microtipped elastomeric stamps as manipulators and built three dimensional silicon microstructures [1]. Silicon units of varied shapes were fabricated in a suspended format on donors, retrieved, delivered, and placed on a target location on a receiver using microtipped stamps. Annealing of the assembled silicon units permanently bound them and completed the micro-masonry procedure

Emergent Spacetime and The Cosmological Constant

We address issues on the origin of gravity and the dark energy (or the cosmological constant) from the perspectives of emergent gravity. We discuss how the emergent gravity reveals a noble, radically different picture about the origin of spacetime, which is crucial for a tenable solution of the cosmological constant problem. In particular, the emergent gravity naturally explains the dynamical origin of flat spacetime, which is absent in Einstein gravity.Comment: 2 pages; Contribution to Proceedings of the workshop "Progress of String Theory and Quantum Field Theory", Osaka City University, Japan, 7-10 December 200