Bounds on the volume fraction of 2-phase, 2-dimensional elastic bodies and on (stress, strain) pairs in composites

Bounds are obtained on the volume fraction in a two-dimensional body containing two elastically isotropic materials with known bulk and shear moduli. These bounds use information about the average stress and strain fields, energy, determinant of the stress, and determinant of the displacement gradient, which can be determined from measurements of the traction and displacement at the boundary. The bounds are sharp if in each phase certain displacement field components are constant. The inequalities we obtain also directly give bounds on the possible (average stress, average strain) pairs in a two-phase, two-dimensional, periodic or statistically homogeneous compositeComment: 16 pages, 2 figures, Submitted to Comptes Rendus Mecaniqu

Localized and complete resonance in plasmonic structures

This paper studies a possible connection between the way the time averaged electromagnetic power dissipated into heat blows up and the anomalous localized resonance in plasmonic structures. We show that there is a setting in which the localized resonance takes place whenever the resonance does and moreover, the power is always bounded and might go to $0$. We also provide another setting in which the resonance is complete and the power goes to infinity whenever resonance occurs; as a consequence of this fact there is no localized resonance. This work is motivated from recent works on cloaking via anomalous localized resonance

Reconstruction and stability in acousto-optic imaging for absorption maps with bounded variation

The aim of this paper is to propose for the first time a reconstruction scheme and a stability result for recovering from acoustic-optic data absorption distributions with bounded variation. The paper extends earlier results on smooth absorption distributions. It opens a door for a mathematical and numerical framework for imaging, from internal data, parameter distributions with high contrast in biological tissues

Doctor of Philosophy

dissertationThis dissertation is concerned with the existence of solutions to fully nonlinear elliptic equations of the form Au = Fu, where A is a differential operator acting on a subspace of the Sobolev space W1,p loc (?), p > 1, ? is a bounded domain in RN and F is an operator depending on lower order terms which also satisfies certain growth conditions. In our study, we use variational methods, fixed point theorems and, especially, sub-supersolution theorems. Our sub-supersolution theorems obtained are motivated by and are more general than those of Vy Le and Schmitt. With our approach, the operator F is allowed to be singular, to contain convection terms and to involve nonlocal terms

Directed hypergraph neural network

To deal with irregular data structure, graph convolution neural networks have been developed by a lot of data scientists. However, data scientists just have concentrated primarily on developing deep neural network method for un-directed graph. In this paper, we will present the novel neural network method for directed hypergraph. In the other words, we will develop not only the novel directed hypergraph neural network method but also the novel directed hypergraph based semi-supervised learning method. These methods are employed to solve the node classification task. The two datasets that are used in the experiments are the cora and the citeseer datasets. Among the classic directed graph based semi-supervised learning method, the novel directed hypergraph based semi-supervised learning method, the novel directed hypergraph neural network method that are utilized to solve this node classification task, we recognize that the novel directed hypergraph neural network achieves the highest accuracies