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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 multiscaleExpand
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 possessExpand
Numerical validation of blow-up solutions of ordinary differential equations
We present a method for validating blow-up solutions of ordinary differential equations (ODEs), which is based on compactifications and the Lyapunov function validation method. Expand
A note on the spectral mapping theorem of quantum walk models
We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular,Expand
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 $x' =Expand
On Blow-Up Solutions of Differential Equations with Poincaré-Type Compactifications
  • K. Matsue
  • Computer Science, Mathematics
  • SIAM J. Appl. Dyn. Syst.
  • 11 September 2018
We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations with quasi-homogeneous desingularization (blowing-up) of singularities and compac... Expand
Quantum walks on simplicial complexes
We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. Expand
Quaternionic quantum walks of Szegedy type and zeta functions of graphs
We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties by using complex right eigenvalues of the corresponding doubly weighted matrix. Expand
On the construction of Lyapunov functions with computer assistance
We provide a systematic methodology for constructing explicit ranges where quadratic Lyapunov functions exist in two stages; negative definiteness of associating matrices and direct approach. Expand
Description of Medium-Range Order in Amorphous Structures by Persistent Homology
The description of amorphous structures has been a long-standing problem, and conventional methods are insufficient for revealing intrinsic structures. In this Letter, we propose a computationalExpand