Probability distributions and weak limit theorems of quaternionic quantum walks in one dimension

The discrete-time quantum walk (QW) is determined by a unitary matrix whose component is complex number. Konno (2015) extended the QW to a walk whose component is quaternion.We call this model quaternionic quantum walk (QQW). The probability distribution of a class of QQWs is the same as that of the QW. On the other hand, a numerical simulation suggests that the probability distribution of a QQW is different from the QW. In this paper, we clarify the difference between the QQW and the QW by weak limit theorems for a class of QQWs.Comment: 11 pages, 2 figures, Interdisciplinary Information Sciences (in press

Noncommutative zeta functions of graphs and their applications

研究成果の概要 (和文) : 本研究では，グラフに付随して定まる有向辺に，非可換な量，すなわち行列や四元数のように積の順序を逆にすると結果が異なるような量，で重みづけした重み付きグラフに対するゼータ関数を定め，非可換な量を成分にもつ行列に対する行列式（非可換行列式）などによる表現の導出など，その主要な性質の解明と関連領域への応用を行った。グラフの第１種重み付きゼータ関数と第２種重み付きゼータ関数の二つについて，これまでの理論を重みが四元数の場合へ一般化し，その成果をグラフ上の四元数量子ウォークのスペクトルの解析に応用した。また，第１種重み付きゼータ関数は，四元数を特殊な場合として含む，より一般の場合への拡張を行った。研究成果の概要 (英文) : We defined several classes of weighted zeta functions of noncommutative weighted graphs; they are considered to have symmetric directed edges that are weighted by noncommutative quantities such as matrices or quaternions. We obtained main properties of the zeta functions such as determinant expressions. We generalized the theories of first and second weighted zeta functions of graphs to the case of quaternion-weighted graphs and applied them to the analysis of the spectra for quaternionic quantum walks on graphs. We also generalized the theory of first weighted zeta functions to much more general situation that includes the case of quaternions

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282