Crystal image analysis using 2D2D synchrosqueezed transforms

We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of non-overlapping segments of 2D general intrinsic mode type functions, which are superpositions of non-linear wave-like components. In particular, crystal defects are interpreted as the irregularity of local energy; crystal rotations are described as the angle deviation of local wave vectors from their references; the gradient of a crystal elastic deformation can be obtained by a linear system generated by local wave vectors. Several numerical examples of synthetic and real crystal images are provided to illustrate the efficiency, robustness, and reliability of our methods.Comment: 27 pages, 17 figure

Foundry: Hierarchical Material Design for Multi-Material Fabrication

We demonstrate a new approach for designing functional material definitions for multi-material fabrication using our system called Foundry. Foundry provides an interactive and visual process for hierarchically designing spatially-varying material properties (e.g., appearance, mechanical, optical). The resulting meta-materials exhibit structure at the micro and macro level and can surpass the qualities of traditional composites. The material definitions are created by composing a set of operators into an operator graph. Each operator performs a volume decomposition operation, remaps space, or constructs and assigns a material composition. The operators are implemented using a domain-specific language for multi-material fabrication; users can easily extend the library by writing their own operators. Foundry can be used to build operator graphs that describe complex, parameterized, resolution-independent, and reusable material definitions. We also describe how to stage the evaluation of the final material definition which in conjunction with progressive refinement, allows for interactive material evaluation even for complex designs. We show sophisticated and functional parts designed with our system.National Science Foundation (U.S.) (1138967)National Science Foundation (U.S.) (1409310)National Science Foundation (U.S.) (1547088)National Science Foundation (U.S.). Graduate Research Fellowship ProgramMassachusetts Institute of Technology. Undergraduate Research Opportunities Progra

About Lorentz invariance in a discrete quantum setting

A common misconception is that Lorentz invariance is inconsistent with a discrete spacetime structure and a minimal length: under Lorentz contraction, a Planck length ruler would be seen as smaller by a boosted observer. We argue that in the context of quantum gravity, the distance between two points becomes an operator and show through a toy model, inspired by Loop Quantum Gravity, that the notion of a quantum of geometry and of discrete spectra of geometric operators, is not inconsistent with Lorentz invariance. The main feature of the model is that a state of definite length for a given observer turns into a superposition of eigenstates of the length operator when seen by a boosted observer. More generally, we discuss the issue of actually measuring distances taking into account the limitations imposed by quantum gravity considerations and we analyze the notion of distance and the phenomenon of Lorentz contraction in the framework of ``deformed (or doubly) special relativity'' (DSR), which tentatively provides an effective description of quantum gravity around a flat background. In order to do this we study the Hilbert space structure of DSR, and study various quantum geometric operators acting on it and analyze their spectral properties. We also discuss the notion of spacetime point in DSR in terms of coherent states. We show how the way Lorentz invariance is preserved in this context is analogous to that in the toy model.Comment: 25 pages, RevTe