2019年度年会(於:東京工業大学)

SUMMARY: As one of the Millennium Prize Problems, the problem of existence and smoothness of the Navier–Stokes equation draws the attention of mathematician from the world. Meanwhile, the verified computing with assistance of computers has proved to be a promising approach to investigate the solution existence to nonlinear equation systems.

In this talk, I will report the latest progress about the solution verification for the stationary Navier–Stokes equation over a non-convex 3D domain. The verification is under the frame of Newton–Kantorovich’s theorem along with the quantitative error analysis for the finite element methods. For the kernel problems in applying Newton–Kantorovich’s theorem, the following schemes are utilized.

1) To bound the norm of the inverse of a differential operator, the algorithm based on the fixed-point theorem (Nakao, 1999) is utilized.

2) To give the a priori error estimation of the projection from solution existing space to finite element spaces, the hypercircle method (Liu–Oishi, 2013) is adopted.

3) The rigorous eigenvalue estimation for differential operators in 3D domain is provided by using the non-conforming finite element method (Liu, 2015).

SUMMARY: Current studies of viral replication deliver detailed time courses of several virological variables, like the amount of virus and the number of target cells, measured over several days of the experiment. Each of these time points provides a snapshot of the virus infection kinetics and is brought about by the complex interplay of target cell infection, viral production and death. It remains a challenge to interpret this data quantitatively and reveal the kinetics of these underlying processes to understand how the viral infection depends on these kinetic properties. In order to decompose the kinetics of virus infection, I introduce a method to “quantitatively” describe the virus infection, and discuss the potential of the combinational analyses with experimental and computational virology.

SUMMARY: When we consider some properties for graphs, forbidden subgraph conditions are frequently used as essential (sufficient) conditions. For example, Duffus, Gould and Jacobson (1981) proved that every 2-connected \(\{K_{1,3},N\}\)-free graph has a Hamiltonian cycle, and Bedrossian (1991) proved that every 2-connected \(\{K_{1,3},B_{1,2}\}\)-free graph has a Hamiltonian cycle, where \(K_{1,3}\) is the star with three leaves, \(N\) is the graph with degree sequence \((3,3,3,1,1,1)\) and \(B_{1,2}\) is the graph with degree sequence \((3,3,2,2,1,1)\) having a triangle. Although above two results were independently given, their forbidden subgraph conditions seem to be similar. Indeed, the speaker and Shoichi Tsuchiya explicitly characterized the connected \(\{K_{1,3},B_{1,2}\}\)-free but not \(N\)-free graphs. Such a characterization, together with Duffus–Gould–Jacobson theorem, leads to Bedrossian’s theorem as a corollary. Thus in forbidden subgraph problem, it is important to discuss about the difference of classes of graphs generated by forbidden subgraph conditions. In this talk, we survey recent progress from this point of view. We also focus on forbidden sugraph conditions generating a finite set of high-connected graphs.

SUMMARY: We extend the complex Newton method. We give the followings for the extended complex Newton method. A relationship between the extended complex Newton method and the Riemann surface. The distribution of roots of extended complex Newton method for \(z^3=1\). Sets of initial values converging at the roots of the extended complex Newton method for \(z^3=1\).

SUMMARY: We are concerned with the numerical evaluation problem of the noncausal stochastic integral with respect to Brownian motion \(W_t(\omega )\) of a random function \(f(t,\omega )\) ;  \(I(f)=\int _0^T f(t,\omega )d_*W_t\). More precisely we intend to construct, on the basis of finite number of observation data \(\{f(t_k,\omega ), 0=t_0<t_1<\cdots <t_n=T\}\), a numerical scheme whose precision level can be much better than \(O(\frac {1}{\sqrt {n}})\). For this purpose we restrict our attention to such a special case where the integrand is of the form \(f(t,\omega ) =g(W_t(\omega ))\) with unknown function \(g(x)\). We develop the discussion in the framework of the noncausal theory of stochastic calculus. The aim of the talk is to present a simple but rapide numerical scheme for the stochastic integral.

SUMMARY: The \(\beta \)-transformation \(T_{\beta }\) and the linear mod 1 transformation \(T_{\beta ,\alpha }\) are transformations on [0,1) defined by \(T_{\beta }(t)=\beta t-\lfloor \beta t\rfloor \) and \(T_{\beta ,\alpha }(t)=\beta t+\alpha -\lfloor \beta t+\alpha \rfloor \)   \((\beta >1\), \(0<\alpha <1)\). We consider how fast the distribution of \(T^{k}_{\beta }([0,1))\) and \(T^{k}_{\beta ,\alpha }([0,1))\) approaches to its invariant density, and give explicit rate of convergence to invariant density function using \(\beta \) or \(\beta \) and \(\alpha \). The proof is proceeded by counting the number of same kind of lines which appear in the graph of \(T^{k}_{\beta }([0,1))\) or \(T^{k}_{\beta ,\alpha }([0,1))\). The base of proof is to show that the ratio of two numbers (or a sum of some numbers) obtained above approaches to \(\beta ^{-j}\) as \(k\rightarrow \infty \).

SUMMARY: Liesegang Phenomena is a kind of typical “Reaction-Diffusion” system with precipitation of resulting stuff of the chemical reaction. When this precipitation occurs, the density of the stuff goes over a certain threshold value, which is called “Saturation Concentration”. The precipitation happens suddenly, if the density reaches the threshold value. This is because the nonlinear term of the system of partial differential equations has a kind of “jump”, namely discontinuity. We consider that the rate of density variation can take whole value of the width of “jump”. We represent it by use of subdifferential notion. Briefly I introduce our work with Prof.s M. Mimira, Rein V. D. Hout, and D. Hilhorst.

SUMMARY: In this talk, we present the way to determine parameters arising in our model equation by a technique of image segmentation for images taken from a movie of smoldering phenomena of a sheet of paper near floor. Our model equation is an interfacial evolution equation and the normal velocity is a linear combination of a constant, the curvature and its second derivative w.r.to the arc-length. The evolution equation is equivalent to the so-called Kuramoto–Sivashinsky equation for a graph. Unknown parameters are the constant and a coefficient of the curvature, and they will be determined by our technique and compared with the theoretical values.

SUMMARY: The Kuramoto–Sivashinsky equation on a Jordan curve is studied from the view point of local bifurcation analysis and image processing. Recently, it was reported that this model equation is valid for not only propagating gaseous flame fronts but also expanding smoldering fronts over thin solids. In this talk, we focus our attention on the instability at a circle solution, and derive a normal form on center manifold. As a result, we see that a rotation wave bifurcates from a circle solution. The existence of this solution implies that rotating effect is inherent in the smoldering combustion phenomena.

SUMMARY: We consider the Lagrange–Galerkin scheme for the Navier–Stokes problem. Because the scheme includes integration of a composite function, it is difficult to implement the scheme exactly. Therefore, the scheme with numerical quadrature is often used in practice. However, the convergence has not been shown. We observed that numerical results of the scheme show instability depending on the time increment. On the other hand, we also observed that the stability is recovered when small time increments are used. Herein we explain the reason. We consider the scheme with quadrature when quadrature points are inside the element and the time increment is sufficiently small. We show results of convergence of the scheme including \(L^2\)-estimate of the velocity, and show numerical results that reflect the theoretical result.

SUMMARY: In this talk, we propose a new numerical scheme for total variation flows whose values are constrained in a Riemannian manifold. Moreover, we prove convergence of time-discretized solution generated by the proposed scheme when initial datum is a piecewise constant function. Finally, we perform numerical results via the proposed scheme.

SUMMARY: Shape optimization requires shape sensitivity analysis, optimization, numerical calculation method. In elasticity, we have built a systematic method using the generalization of J-integral (GJ-integral) for shape sensitivity analysis, \(H^1\)-gradient method for optimization and finite element method for numerical calculation method. The feature of this method is a general-purpose method that is applicable even if the solution has singularity. In Stokes problem, we cannot use the method stated just above because of incompressible. We obtain the shape sensitivity analysis for Lagrangian, and derive GJ-integral in Stokes problem. Examples of the calculation will be shown in our talk.

SUMMARY: We characterize some steady solutions of the Euler–Arnold equations and Navier–Stokes–Taylor equations by geometric structures of a surface. We first investigate physical aspects of a hydrodynamic Killing vector field as a steady solution of the Euler–Arnold equations and the Navier–Stokes–Taylor equations from the conformal structure and the singular Riemannian foliated structure generated by the Killing vector field. We compare physical properties of a potential vector field with that of the Killing vector field. Deriving exact solutions and steady solutions with singular vorticities and exact infinitesimal flux, we discuss them in terms of geometric structures of a surface.

SUMMARY: The harmonic measure is closely related to the Dirichlet boundary value problem of the Laplace equation. Moreover, it appears in probability theory and harmonic analysis, so the study of the harmonic measure is recognized to be an important topic. It is desirable to obtain an analytic form of the harmonic measure, however, it is impossible in general, so numerical study also plays a key role. In this talk, we construct a numerical scheme for computing harmonic measure on a closed surface based on the method of fundamental solutions and compare numerical results with analytic ones.

SUMMARY: We use a simple model of the dynamics of an inviscid vortex sheet desribed by the Birkhoff–Rott equation to gain some fundamental insights about the potential for stabilization of shear layers using feedback control. Let us consider two arrays of point sinks/sources located a certain distance above and below the vortex sheet as actuators subject to the mass conservation. We demonstrate using analytical computations that the Birkhoff–Rott equation linearized around the flat-sheet configuration is in principle controllable. Second, we design a state-based LQR stabilization strategy with using high precision arithmetics.

SUMMARY: The algebraic stability theorem is an important part of the stability theorem in the theory of persistent homology and guarantees that the persistence diagram is robust to changes in the given persistence module. Our motivation is to derive the algebraic stability theorem for the zigzag persistence module. It is not easy to prove it directly. It is known that the derived category of the persistence module and the zigzag one are equivalent. Thus our strategy is to derive the algebraic stability theorem for the zigzag persistence module from the ordinary one by using the derived category. In this talk, we will discuss the algebraic stability theorem for the derived category of the persistence module.

SUMMARY: Percolation theory is a branch of probability theory which describes the behavior of clusters in a random graph. Recently, craze formation in polymer materials is gaining attention as a new type of percolation phenomenon in the sense that a large void corresponding to a craze of the polymer starts to appear by the process of coalescence of many small voids. In this talk, I introduce a new percolation model motivated from the craze formation of polymer materials. For the sake of modeling the coalescence of nanovoids, this model focuses on clusters of holes, which are formulated as homology generators in codimension one, while the classical percolation theory mainly studies clusters of vertices (i.e., 0-dimensional objects).

SUMMARY: A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules by using their restrictions onto lines, giving a family of 1D persistence modules. We explore what 1D persistence modules can be obtained as a restriction of indecomposable 2D persistence modules to a single line. To this end, we give a constructive proof that any 1D persistence module can in fact be found as a restriction of some indecomposable 2D persistence module. As another consequence of our construction, we are able to exhibit indecomposable 2D persistence modules whose support has holes.

SUMMARY: In persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. In multidimensional persistence it is known to be impossible to classify all indecomposable modules. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show that over the “equioriented” commutative \(2\)D grid, these concepts are equivalent. Moreover, we provide an algorithm for determining whether or not an \(n\)D persistence module is interval/pre-interval/thin-decomposable, under certain finiteness conditions and without explicitly computing decompositions.

SUMMARY: We propose a discrete theory and a construction algorithm of Reeb graphs in which finite 2D data are sampled from real-valued functions. Owing to their natural derivation, we can bridge a gap between discreteness and continuity in the Reeb graph problem. Since the theory is based on so-called merge trees of 0-th persistent homology for sublevelset and superlevelset filtrations, stability of the algorithm is to be expected. We also show an application to Yokoyama’s tree-representation theory for topological fluid dynamics in which we have one-to-one correspondence between labeled trees and 2D Hamiltonian flows under topological classification. Although tree-representations are computed by hand in preceding studies, we have established a conversion algorithm using our Reeb graph theory.

SUMMARY: In the case of finite element approximation for PDEs in a smooth domain, we calculate numerical solution in a polygonal domain approximating the original domain. Then, it may occur that we calculate approximate solution of another problem. In particular, we need to be more careful with boundary condition including derivatives like reduced-FSI model. For standard FEM, there are many study for numerical calculation with several boundary conditions in a smooth domain, but few studies exist for another method like discontinuous Galerkin method. In this study, we show the analysis and some numerical results of discontinuous Galerkin method for Poisson equations with a Robin boundary condition in a smooth domain.

SUMMARY: We consider the planer three-body problem with generalized force. Some non-integrability results for these systems have been obtained by analyzing the variational equations along the homothetic solutions. But we can not apply it to several exceptional cases. For example, when the system has inverse-square potentials the variational equations along the homothetic solutions are always solvable. We obtain sufficiently conditions for non-integrability for these exceptional cases by focusing on some particular solutions that are different from homothetic solutions.

SUMMARY: We derive the second order perturbation equations of the Einstein equations in Minkowski background metric. To make simulations with the perturbations equations, we derive the Hamiltonian formualtion of the equations. In addition, we propose a numerical scheme of the equations for calculating precise simulations.

SUMMARY: In this talk, we consider a numerical method for proving the existence of solutions for the complex Ginzburg–Landau equations. Our verification principle is based on the simplified Newton operator for time-dependent Fourier coefficients. We derive a sufficient condition for verifying the simplified Newton operator becomes the contraction mapping on a neighborhood of a numerically computed approximate solution, which is given by Chebyshev–Fourier spectral methods.

SUMMARY: A higher order error estimation for the approximate solution of the Poisson equation is presented. The proposed procedure is able to be applicable to residual iteration techniques for the verification of solutions to nonlinear elliptic equations. Some numerical examples by finite element method comparing other approaches will be shown.

SUMMARY: The east profile of Mt. Fuji has a middle part which is approximately represented by the logarithmic curve (Milne, 1878). In this talk we shall show that the east surface of Mt. Fuji and the whole surface of Mt. Kaimondake have respectively their middle parts which are approximately represented by the rotational surfaces around their vertical axes through points of their summits and that their generating curves are the logarithmic curves, by investigating their contour maps with scale of 1:25,000 published by Japan geological agency. Therefore we may observe on these surfaces the logarithmic potential property.

SUMMARY: Motion of grain boundaries with dynamic lattice orientations and with triple junction drag is considered. We derive some evolution equations ensuring dissipation of the grain boundary energy via the energetic variational approach. We take the relaxation limit to the curvature effects on the equations, to take into account of the effect of the dynamic lattice orientations and the triple junction drag, and show the solvability of the relaxation equations.

SUMMARY: The dynamics of a traveling pulse solution arising in a bistable three-component reaction-diffusion system is considered both numerically and analytically. Depending on the parameter values, the traveling pulse exhibits a variety of behavior when it encounters a jump-type spatial heterogeneity. To analytically elucidate its mechanism, four dimensional ODEs are derived by means of multiple scales method, which capture the essential features of the pulse motion observed for the original PDE system. In the talk, we present the numerical results of the heterogeneity-induced pulse behavior, and utilize the reduced ODEs to clarify the mechanism for the pulse dynamics from a viewpoint of bifurcation theory.

SUMMARY: We propose a simple two dimensional snow crystal growth model and discuss about its solvability. The model is based on a generalized hexagonal crystalline motion with several singularities such as facet collision, facet merging, facet generation, and facet breaking. It also includes the effect of diffusion of supersaturated vapor and the Gibbs–Thomson law. We prove unique existence of the solution locally in time and give a simple formula to obtain the facet velocity by means of the single layer potential on the facets. In our numerical examples, setting a suitable critical length of the facet, we can observe typical dendritic crystal growth. This work is based on the collaboration with Ryohei Yamaoka, Kanazawa University and his master’s thesis.

SUMMARY: In this study, a kind of generalization is proposed for an integer-type algorithm for solving higher-order linear ODEs, which was proposed by the author and M. Hayashi several years ago, by means of algebraic extensions of the field of rational functions. This integer-type algorithm, which can solve ODEs only by means of four arithmetic operations among integers, was widely applicable for the higher-order linear ODEs with rational coefficient functions over Q (rational numbers). However, we can widen the range of its applications to the cases where the coefficient functions belong to algebraic extensions of the field of rational functions over Q. For example, some successful numerical examples are given for the Schrödinger equations whose potential functions belong to the simple extension of the field of rational functions which is obtained by adjoining the square root of a positive-valued rational function.

SUMMARY: Determination of whether periodic orbits, homoclinic orbits, first integrals or commutative vector fields may persist under perturbations is one of the most important problems in the field of dynamical systems. In this talk, we give several theorems on necessary conditions for their persistence in general perturbed systems. Moreover, we consider periodic perturbations of one-degree-of-freedom Hamiltonian systems and describe some relationships between our results and the standard Melnikov method for periodic orbits and homoclinic orbits.

SUMMARY: We consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. We extend the Melnikov method to reversible systems and obtain theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system.

SUMMARY: In this talk, we study the asymptotic behavior of an infection age structured SIR epidemic model with spatial diffusion in the case of Neumann boundary condition. By using the method of characteristics, we transform the model into a system of a reaction-diffusion equation and an integral equation of Volterra type. We then define the basic reproduction number \(R_0\) and show that if \(R_0 < 1\), then the disease-free steady state is globally attractive, whereas if \(R_0 > 1\), then the disease is persistent. Moreover, under an additional assumption that the maximum age of infectiousness is finite, we show that if \(R_0 > 1\), then the constant endemic steady state is globally attractive.

SUMMARY: One of the most important functions of the epidermis is the functional barrier. In this study, we focus on two barrier functions: one is the stratum corneum (SC) that consists of cornified cells and inter-cellular lipids, and the other is the tight junctions (TJs) appearing in the stratum granulosum (SG). The mechanism of the occurrence and maintenance of TJs remains unexplained. Using a mathematical model of the epidermis, we propose a mechanism of the stable formation of TJs. We also evaluate the barrier function in the SC with a mathematical modeling for epidermal desquamation.

SUMMARY: Vegetation patterns and its environmental conditions have a close relationship. The Daisyworld model which is one of the conceptual earth system model introduced by Watson and Lovelock in 1983 may give us new viewpoints about the relationship. In this talk, we consider two-dimensional Daisyworld model to which is extended one-dimensional one of Adams, Carr, Lenton and White in 2003. In our model, the white and black daisies form the three principal types of segregation patterns depending on the intensity of solar luminosity. The purpose of this talk is to discuss in terms of mathematics and biologic that Turing’s mechanisms are inherent in these pattern formations.

SUMMARY: Bacillus subtilis uses different cell types to suit environmental conditions and cell density. The subpopulation of each cell type exhibits various environment-sensitive properties. Furthermore, division of labor among the subpopulations results in flexible development for the community as a whole. Here we present a simple mathematical model of cell type regulation of B. subtilis. We report a necessary and sufficient condition for hysteresis of cell type selection in the model, and discuss how the cell state dynamics is controlled in response to environmental variation.

SUMMARY: A tournament \(T\) is an orientation of a complete graph. Here we consider tournaments with vertices labelled by \(\{1,2,\ldots \}\). For a permutation \(\pi \) on vertices, an arc \((x,y)\) of \(T\) is called consistent if \(x\) precedes \(y\) in \(\pi \). Erdős–Moon (1965) mentioned the problem finding explicit constructions of tournaments with a small number of consistent arcs. Alon–Spencer (2000) found that Paley tournaments are such tournaments. In this talk, we give many such explicit tournaments of more flexible orders. Our method is based on a digraph-version of the expander-mixing lemma found by Vu (2008). Moreover our discussion provides a wide generalization of Alon–Spencer’s proof.

SUMMARY: We use triple intersection numbers of association schemes to show non-existence of tight \(4\)-designs in Hamming association schemes \(H(n,6)\). Combining with a result by Noda (1979), this completes the classification of tight \(4\)-designs in \(H(n,q)\).

SUMMARY: We give a classification of all matrices \(C_1,C_2\in \mathbb {F}_2^{\mathbb {N}\times \mathbb {N}}\) which generate a digital \((0,2)\)-sequence in base \(2\). This gives us an implication for Markov-chain quasi Monte-Carlo point sets.

SUMMARY: In this talk we will give a generalization of the hook formula for the generating function of reverse plane partitions on \(d\)-complete posets to skew \(d\)-complete posets using Schubert calculus of equivariant \(K\)-theory. This gives an alternative uniform proof for the \(d\)-complete posets.

SUMMARY: In this talk, we introduce reflexive polytopes \(\mathcal {B}_G\) arising from bipartitle graphs \(G\), and discuss their \(\delta \)-polynomials. Since \(\mathcal {B}_G\) has a regular unimodular triangulation, its \(\delta \)-polynomial is palindromic and unimodal. We show stronger properties for the \(\delta \)-polynomial of \(\mathcal {B}_G\). In fact, \(\delta \)-polynomial of \(\mathcal {B}_G\) is \(\gamma \)-positive and its \(\gamma \)-polynomial is given by an interior polynomial (a version of Tutte polynomial of a hypergraph). Moreover, the \(\delta \)-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted.

SUMMARY: Most of the tree enumeration formulas are generating functions or recurrence formulas. In this talk, we show the explicit formula for the number of unlabeled rooted trees with a certain condition. The formula is described in terms of Young tableaux.

SUMMARY: Let us consider the forests with \(k\) components in the complete graph. We define \(\Phi \) to be the weighted generating function for them. We calculate the eigenvalues of the Hessian matrix of \(\Phi \) to show that the Hessian of \(\Phi \) does not vanish.

SUMMARY: It is known that we can continuously flatten the surface of a regular tetrahedron onto any of its faces by moving creases to change the shapes of some faces successively. Let \(P_n\) be an \(n\)-dimensional regular simplex with \(n \ge 4\), and \(S\) be the set of its 2-dimensional faces, in other words, the 2-dimensional skeleton of the triangular faces in \(P_n\). We show that \(S\) can be continuously flattened onto any face \(F\) of \(S\) such that at least two thirds of the edges and two ninths of the triangular faces are rigid during the motion.

SUMMARY: We consider the weighted Kirchhoff index of a graph \(G\), and present a generalization of Somodi’s Theorem on one of the Kirchhoff index of a graph. Furthermore, we give an explicit formula for the weighted Kirchhoff index of a regular covering of \(G\) in terms of that of \(G\).

SUMMARY: For a given graph, we consider an optimization problem in which we explore an orientation that minimize an objective function whose value is determined by the out-degrees of the vertices under the orientation. When the orientation is restricted to acyclic ones, such a problem is related to the recognition of shellability of simplicial complexes, so the optimization problem is considered to be hard. In this talk we show that, without acyclicity constraint, the optimization problem under consideration can be solved in polynomial time. This result indicates that the hardness of shellability recognition is caused by the acyclicity.

SUMMARY: An edge of \(k\)-connected graph is said to be \(k\)-contractible if the contraction of it results in a \(k\)-connected graph. A condition on the subgraph induced by the neighborhood of each vertex of a \(k\)-connected graph said to be a local condition of the graph. We present three local conditions for a \(k\)-connected graph to have a \(k\)-contractible edge.

SUMMARY: Some recent results on the optimal proper connection number in connected graphs will be reviewed.

SUMMARY: A matching \(M\) in a graph \(G\) is extendable if there exists a perfect of \(G\) containing \(M\). Also, \(M\) is a distance \(d\) matching if the distance of every pair of distinct edges in \(M\) is at least \(d\). A graph \(G\) is distance \(d\) matchable if every distance \(d\) matching in \(G\) is extendable, regardless of its size. In this talk, we discuss the distance \(d\) matchability of star-free graphs. In particular, we report that for every integer \(k\ge 3\), there exists an integer \(d\) such that every locally \((k-1)\)-connected \(K_{1,k}\)-free graph of even order is distance \(d\) matchable.

SUMMARY: In this talk, we consider triangulations of the plane. Ozeki and Zamfirescu asked whether there are non-hamiltonian 1-tough triangulations in which every two separating triangles are disjoint. We answer this question in the affirmative by proving that there are infinitely many non-hamiltonian 1-tough triangulations with pairwise disjoint separating triangles.

SUMMARY: It is well-known that any Eulerian plane graph \(G\) is face \(2\)-colorable and admits an orientation, which is an assignment of a direction to each edge of \(G\), such that incoming edges and outgoing edges appear alternately around any \(v \in V(G)\); we say that such a vertex \(v\) has the alternate property, and that such an orientation is good. In this talk, we discuss orientations given to Eulerian plane graphs such that some specified vertices do not have the alternate property, and give a characterization in terms of the radial graph of the graph. Furthermore, for a given properly drawn graph on the plane (with crossing points), we discuss whether it has a good orientation or not.

SUMMARY: The beans function \(B_G(x)\) of a connected graph \(G\) is defined as the maximum number of points on \(G\) such that any pair of points have distance at least \(x>0\). We give a method of constructing non-isomorphic graphs with the same beans function.

SUMMARY: A triangulation on a closed surface is a graph embedded on the surface such that every face is bounded by a cycle of length \(3\). Let \(G\) be a triangulation on a closed surface and \(n \ge 3\) be a natural number. A coloring \(c : V(G) \to \mathbb {Z}_n\) is called an \(n\)-triad coloring if \(\{c(u), c(v), c(w)\}\) belongs to \(\{\{i, i+1, i+2\} \mid i \in \mathbb {Z}_n\}\) for any face \(uvw\) of \(G\).

We would like to determine the set of integers \(n\) such that \(G\) has \(n\)-triad colorings. In this talk, to determine such a set completely, we shall introduce an algorithm for enumerating triad colorings of a given triangulation \(G\) by using perfect matchings of a dual of \(G\).

SUMMARY: Localization and ballistic spreading are characteristic properties of quantum walks with contrast to random walks. We consider mainly localization of space-homogeneous discrete time quantum walks on the \(d\)-dimensional lattice. A necessary and sufficient condition of localization was presented by Tate in 2014. The stationary measure of the Grover walk on the \(d\)-dimensional lattice was given by Komatsu and Konno in 2017. Here we obtain a necessary and sufficient condition of localization via the stationary measure. Moreover, we get a proof of non-existence of localization of the Fourier walk on the \(d\)-dimensional lattice with \(2\leq d\leq 5\) by using our result.

SUMMARY: Quantum walks determined by the coin operator on graphs have been intensively studied. The typical examples of coin operator are the Grover and Fourier matrices. The periodicity of the Grover walk is well investigated. However, the corresponding result on the Fourier walk is not known. In this talk, we consider the Fourier walk on graphs whose degree of vertex is power of a prime number. Then, we present a necessary condition for the construction of graphs to have the finite period. As an application of our result, we show that the Fourier walks do not have any finite period for some classes of graphs such as Hamming graphs including hyper cubes, and Wheel graphs.

SUMMARY: Quantum walks are quantum mechanical counterparts of random walks and promising platforms expected to realize topological phenomena. Here we consider two-phase split-step quantum walks on cycles defined by Balu et al., which have two different coins across the boundary of two regions. In this talk, we analyze the long-time behavior of the split-step quantum walk by using a spectral mapping theorem.

SUMMARY: The orthogonal polynomial is the set of polynomials determined by weight functions. In our study, we take limit density functions of the weak limit theorem for quantum walk on the two-dimensional lattice as a weight function which determines orthogonal polynomial of two variables. Here, we construct the orthogonal polynomial by using the Gram–Schmidt orthonormalization with a monomial order, and also get the three-term relation which is an important property of orthogonal polynomial. Moreover, we present limit density function with one parameter by projecting a two-dimensional quantum walk to one dimension. In order to investigate the orthogonal polynomial of two variables, we consider an orthogonal polynomial determined by these limit density functions.

SUMMARY: Quantum walks are regarded as quantum versions of random walks and are applied to several study fields. The time evolution of the quantum walks is defined by a unitary process. Due to the unitarity, the behavior of quantum walks is quite different from that of random walks. Recently, as a special example of such a difference, periodicity of quantum walks is studied. Our ultimate aim is to characterize spatial structure which yield periodic quantum walks. In particular, such characterization for the Grover walks on graphs is intensively analyzed by using a spectral mapping theorem. In this talk, we provide some necessary conditions of graphs to give an odd-periodic Grover walk through dynamical analysis.

SUMMARY: The \(t\)-tessellation of graph \(G=(V,E)\) is a sequence of \(t\)-kinds of clique decompositions so that all the edges of \(E(G)\) are covered by these decompositions. We introduce a directed graph from some abelian covering graphs so that the indegree and outdegree are the same for every vertex. Since this deformation reduces the degree of each vertex, we can save the dimension of the local coin assigned at each vertex of the quantum walk. This reduction is expected to make it possible for some experimental implementations of quantum walks on graphs, in particular, in the case of degree \(2\). In this talk, we show a connection between the directed graph and \(t\)-tessellation, and give some examples of asymptotic behavior of quantum walks on directed graphs.

SUMMARY: In this talk, we give general expression of the solutions of the eigenvalue problem, and discuss the stationary measure mainly for three-state quantum walk (QW) by using our new recipe. So far, two kinds of limit theorems have described the characteristic properties of QWs mathematically. The one is the limit theorem, which is composed of the time-averaged limit measure, corresponding to localization. The other is the weak convergence theorem, which expresses the ballistic spreading of the walker by the weak limit measure. In recent years, stationary measure for QW has received attention as another key measure for the asymptotic distribution of QW. The stationary measure provides the stationary distribution, for instance. Firstly, we propose a new type of theorems to construct the stationary measure by using transfer matrices, and then, we show some concrete examples comparing the results with that obtained by other methods we had developed. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.

SUMMARY: In this talk, we deal with the period \(T_N\) of the quantum walk with moving shift and flip-flop shift on a cycle \(C_N\) with \(N\) vertices. Konno et al. in 2017 showed that \(T_2=2, T_4=8, T_8=24\) and \(T_N=\infty \) if \(N \not = 2, 4, 8\) for the two-state Hadamard walk with moving shift on \(C_N\) by using the path counting and cyclotomic polynomials. Here we proved that \(T_N=\infty \) if \(N \not =3\) for the three-state Grover walk and Fourier walk with moving shift and flip-flop shift on \(C_N\) by using the ring of integers of the cyclotomic fields and the property of eigenvalues for unitary matrix which determines the evolution of the walk.

SUMMARY: We consider one-dimensional two state quantum walks. We derive a weak limit theorem for a class of long range type quantum walks.

SUMMARY: In this talk, we consider a quantum walk whose time evolution is given by \(U=SC\), where \(S\) and \(C\) are self-adjoint and \(S\) is unitary on an abstract Hilbert space. We emphasize that \(U\) is not always unitary. In the case that \(U\) is unitary, the spectral mapping theorem of quantum walk is provided by Y. Higuchi, E. Segawa and A. Suzuki. By the way, K. Mochizuki, D. Kim and H. Obuse introduce a new model of quantum walk, derived from recent experiment, whose time evolution is not unitary. This model is a concrete example of our model. We provide a spectral mapping theorem for the part of all eigenvalues of \(U\).