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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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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