Kaname Matsue

Geometric Frustration of Icosahedron in Metallic Glasses

Order, Order The structure of glassy materials, which are known to have short-range order but no long-range pattern, continues to be a puzzle. One current theory is that some glassy materials possess icosahedral ordering, a motif that cannot show translational periodicity. Hirata et al. (p. 376, published online 11 July) obtained diffraction patterns from subnanometer volumes in a metallic glass, which show some, but not all, of the expected features of an icosahedron. Simulations suggest that the patterns arise from icosahedrons distorted to include features of the face-centered cubic structure. This observation is different from the predictions of molecular dynamics simulations and provides pivotal information in understanding the competition between the formation of the globally inexpensive long-range order and the locally inexpensive short-range order. Small-volume regions in a bulk metallic glass show icosahedral ordering distorted by partial face-centered cubic symmetry. Icosahedral order has been suggested as the prevalent atomic motif of supercooled liquids and metallic glasses for more than half a century, because the icosahedron is highly close-packed but is difficult to grow, owing to structure frustration and the lack of translational periodicity. By means of angstrom-beam electron diffraction of single icosahedra, we report experimental observation of local icosahedral order in metallic glasses. All the detected icosahedra were found to be distorted with partially face-centered cubic symmetry, presenting compelling evidence on geometric frustration of local icosahedral order in metallic glasses.

Hierarchical structures of amorphous solids characterized by persistent homology

Significance Persistent homology is an emerging mathematical concept for characterizing shapes of data. In particular, it provides a tool called the persistence diagram that extracts multiscale topological features such as rings and cavities embedded in atomic configurations. This article presents a unified method using persistence diagrams for studying the geometry of atomic configurations in amorphous solids. The method highlights hierarchical structures that conventional techniques could not have treated appropriately. This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.

Numerical validation of blow-up solutions of ordinary differential equations

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach

We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system

Quantum walks on simplicial complexes

x = f(x,y,epsilon), y = epsilon g(x,y,epsilon)

On the construction of Lyapunov functions with computer assistance

with one-dimensional slow variable

Resonant-tunneling in discrete-time quantum walk

y

Overview of Cubical Homology

. Our validation procedure is based on topological tools called isolating blocks, cone condition and covering relations. Such tools provide us with existence theorems of global orbits which shadow singular orbits in terms of a new concept, the covering-exchange. Additional techniques called slow shadowing and

Application of Computational Homology to Metallic Glass Structures

m

Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media

-cones are also developed. These techniques give us not only generalized topological verification theorems, but also easy implementations for validating trajectories near slow manifolds in a wide range, via rigorous numerics. Our procedure is available to validate global orbits not only for sufficiently small