Search

We present the characteristic polynomial for the transition matrix of a vertex-face walk on a graph, and obtain its spectra. Furthermore, we express the characteristic polynomial for the transition matrix of a vertex-face walk on the 2-dimensional torus by using its adjacency matrix, and obtain its spectra. As an application, we define a new walk-type zeta function with respect to the transition matrix of a vertex-face walk on the 2-dimensional torus, and present its explicit formula., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2103.12971, arXiv:2011.14162

Our previous work delt with the zeta function for the interacting particle system (IPS) including quantum cellular automaton (QCA) as a typical model in the study of ``IPS/Zeta Correspondence". On the other hand, the absolute zeta function is a zeta function over F_1 defined by a function satisfying an absolute automorphy. This paper proves that a new zeta function given by QCA is an absolute automorphic form of weight depending on the size of the configuration space. As an example, we calculate an absolute zeta function for a tensor-type QCA, and show that it is expressed as the multiple gamma function. In addition, we obtain its functional equation by the multiple sine function., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2306.14625

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks. Afterwards we deal with higher-dimensional quantum walks. For comparison with the case of the quantum walk, we also treat the case of higher-dimensional random walks. Our results bridge between the Mahler measure and the zeta function via quantum walks for the first time., Comment: 27 pages. arXiv admin note: text overlap with arXiv:2109.07664, arXiv:2104.10287

We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function and a generalized zeta function of a graph $G$ with respect to its generalized Grover matrix as an analog of the Ihara zeta function and present explicit formulas for their zeta functions for a vertex-transitive graph. As applications, we express the limit on the generalized zeta functions of a family of finite vertex-transitive regular graphs by an integral. Furthermore, we give the limit on the generalized zeta functions of a family of finite tori as an integral expression., Comment: 10 pages. arXiv admin note: text overlap with arXiv:2011.14162

The Ronkin function was defined by Ronkin in the consideration of the zeros of almost periodic function. Recently, this function has been used in various research fields in mathematics, physics and so on. Especially in mathematics, it has a closed connections with tropical geometry, amoebas, Newton polytopes and dimer models. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on Zeta Correspondence. The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Ronkin function and our zeta function for random walks and quantum walks. Firstly we consider this relation in the case of one-dimensional random walks. Afterwards we deal with higher-dimensional random walks. For comparison with the case of the quantum walk, we also treat the case of one-dimensional quantum walks. Our results bridge between the Ronkin function and the zeta function via quantum walks for the first time., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2202.05966

We define a zeta function woth respect to the twisted Grover matrix of a mixed digraph, and present an exponential expression and a determinant expression of this zeta function. As an application, we give a trace formula with respect to the twisted Grover matrix of a mixed digraph., Comment: 17 pages

We define a zeta function of a finite graph derived from time evolution matrix of quantum walk, and give its determinant expression. Furthermore, we generalize the above result to a periodic graph., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1910.12782

In our previous work, we investigated the relation between zeta functions and discrete-time models including random and quantum walks. In this paper, we introduce a zeta function for the continuous-time model (CTM) and consider CTMs including the corresponding random and quantum walks on the d-dimensional torus., Comment: 9 pages, minor corrections, Quantum Studies: Mathematics and Foundations, Volume 9, pp.165-173 (2022)

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak., Comment: The draft is improved: Typos are fixed, Corollaries 4.3 and 4.5 are improved, and Proposition 4.6 and Remark 4.7 are added

Aims: Myostatin (Mstn) is a member of the transforming growth factor-β (TGF-β) family that inhibits muscle differentiation. In this study, we aimed to identify the relationships between Mstn, thyroid hormone receptor alpha (TRα), and myosin heavy chain (MyHC) isoform expression during early postnatal development. Methods: We investigated the expression of Mstn, TRα, and MyHCs (embryonic, slow, IIa, IIb, and IIx) using quantitative real-time RT-PCR and ELISA (Mstn) in postnatal mouse muscles between day 0 and day 10. We also examined the correlations between Mstn, TR and MyHCs during the early development of mouse masseter muscle (MM) and rectus femoris muscle (RFM). Results: Distinct Mstn mRNA expression patterns were observed in the two muscles despite nearly non-significant changes in the Mstn protein abundance in MM. The expression pattern of the TRα mRNA in the MM differed from that observed in the RFM. The expression of MyHC IIa, IIb and IIx mRNAs increased in the MM and decreased in the RFM from day 0 to day 10, whereas embryonic fiber MyHC mRNA expression was similar in both muscle types. Principal component analysis showed the existence of a correlation between: (1) TRα and MyHC, (2) Mstn and MyHC, and (3) TRα and Mstn in MM. The correlations were different in RFM and MM. Cluster analyses identified the distinct clusters: cluster 1, days 0-4 for the MM and day 0 for the RFM;cluster 2, day 6 for the MM and day 2 for the RFM; and cluster 3, days 8-10 for the MM and days 4-10 for the RFM. Conclusions: These data suggest that TRα influences MyHC expression in both muscle types. In addition, Mstn has a limited effect in the MM related to the expression of individual MyHCs, as opposed to its role in the RFM, at early postnatal developmental stages. TRα could be involved in regulating both the temporal expression of MyHCs and Mstn at the early postnatal stages in the MM and RFM.

We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph., Comment: 20 pages

We give a new determinant expression for the characteristic polynomial of the bond scattering matrix of a quantum graph G. Also, we give a decomposition formula for the characteristic polynomial of the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the characteristic polynomial of the bond scattering matrix of a regular covering of G by means of its L-functions. As an application, we introduce three types of quantum graph walks, and treat their relation., Comment: 27 page, 2 figures