Related papers: What is a crystal?
We investigate the interplay of crystal bases and completions in the sense of Enright on certain nonintegrable representations of quantum groups. We define completions of crystal bases, show that this notion of completion is compatible with…
Quasicrystals whose building blocks are of mesoscopic rather than atomic scale have recently been discovered in several soft-matter systems. Contrary to metallurgic quasicrystals whose source of stability remains a question of great debate…
The diffraction pattern of a single non-periodic compact object, such as a molecule, is continuous and is proportional to the square modulus of the Fourier transform of that object. When arrayed in a crystal, the coherent sum of the…
This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular,…
In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. Recently, the method has been applied to quasicrystals and modulated crystals. This paper phrases…
In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why…
The possibility for the occurrence in crystals of a phenomenon, resembling turbulence, is discussed. This phenomenon, called {\it heterophase turbulence}, is manifested by the fluctuational appearance inside a crystalline sample of…
Order and disorder constitute two fundamental and opposite themes in condensed matter physics and materials science. Crystals are considered the epitome of order, characterised by long-range translational order. The discovery of…
Glasses constitute a widespread form of solid matter, and glass production has been an important human technology for more than 3000 years. Despite that long history, new ways to understand the fundamental physics of glasses continue to…
From the days when superconductivity was discovered its science was entangled by the unresolved problem of the relationship between superconductive state, its crystal structure and its phase transitions. The problem was exacerbated by the…
Out of equilibrium states in glasses and crystals have been a major topic of research in condensed-matter physics for many years, and the idea of time crystals has triggered a flurry of new research. Here, we provide the first description…
Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of `complex systems.' In this talk I review some of the basic notions of spin glass physics, and discuss how some of…
A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by…
The prediction of material structure from chemical composition has been a long-standing challenge in natural science. Although there have been various methodological developments and successes with computer simulations, the prediction of…
We propose a mathematical description of crystal structure: underlying translational periodicity together with the distinct atomic positions up to the symmetry operations in the unit cell. It is consistent with the international table of…
The science of complexity is far from being fully understood and even its foundations are not well established. On the other hand, during the last decade, the random motion of particles or waves - the so-called diffusion - has been known…
We find an order-disorder transition from crystals to disordered crystals for static packings of frictionless spheres. While the geometric indicators are mostly blind to the transition, disordered crystals already exhibit properties apart…
For many years, quasicrystals were observed only as solid-state metallic alloys, yet current research is now actively exploring their formation in a variety of soft materials, including systems of macromolecules, nanoparticles and colloids.…
The fundamental issues of symmetry related to chirality are discussed and applied to simple situations relevant to liquid crystals. We show that any chiral measure of a geometric object is a pseudoscalar (invariant under proper rotations…
The large amount of powder diffraction data for which the corresponding crystal structures have not yet been identified suggests the existence of numerous undiscovered, physically relevant crystal structure prototypes. In this paper, we…