2017年度秋季総合分科会(於:山形大学)

SUMMARY: O. Ore (1960) proved that if the degree sum of every pair of non-adjacent vertices is at least the order of the graph, then the graph is hamiltonian. The study of sufficient conditions on degrees for the existence of a Hamilton cycle started from such a classical result. In this talk, as one of the generalizations of Ore’s theorem, we focus on degree sum conditions for the existence of a 2-factor with a specified number of components in general simple graphs, bipartite graphs and directed graphs, and we survey results including recent progress on the research field. We also discuss the difference from the results on Hamilton cycles.

SUMMARY: Validated computations (rigorous numerics) have been applied to dynamical systems, such as the existence of equilibria, periodic orbits, connecting orbits and bifurcations and so on as well as their stability, for a couple of decades. I believe that one of true significance of rigorous numerics to dynamical systems is validation of objects which are very difficult to detect from both mathematical and numerical approach. The root will be either reduction of problems to fixed point problems for nonlinear maps or analysis based on saddle-type equilibria or invariant sets. In this talk, I develop an overview of rigorous numerics to dynamical systems derived from saddles in terms of topological tools such as isolating blocks, cones and Lyapunov functions, as well as various results by analytic approach.

SUMMARY: Delay embedding is well-known for non-linear time-series analysis, and it is used in several research fields such as physics, informatics, neuroscience and so on. The celebrated theorem of Takens ensures validity of the delay embedding analysis: embedded data preserves topological properties, which the original dynamics possesses, if one embeds into some phase space with sufficiently high dimension. This means that, for example, an attractor can be reconstructed by the delay coordinate system topologically. However, configuration of an embedded dataset may easily vary with the delay width and the delay dimension, namely, “the way of embedding”. In a practical sense, this sensitivity may cause degradation of reliability of the method, therefore it is natural to require robustness of the result obtained by the embedding method in certain sense. In this study, we investigate the mathematical structure of the framework of delay-embedding analysis to provide Ansatz to choose the appropriate way of embedding, in order to utilise for time-series prediction. In short, mathematical theories of the Hilbert–Schmidt integral operator and the corresponding Sturm–Liouville eigenvalue problem underlie the framework. Using these mathematical theories, one can derive error estimates of mode decomposition obtained by the present method and can obtain the phase-space reconstruction by using the leading modes of the decomposition. In this talk, we will show some results for some numerical and experimental datasets to validate the present method.

SUMMARY: Ramanujan graph were defined by Lubotzky–Phillips–Sarnak in 1988. It is well known that Ramanujan graphs have nice properties as networks. In combinatorics, to give explicit constructions of Ramanujan graphs is recognized as a very interesting problems. In this talk, we give some explicit constructions of Ramanujan graphs as Cayley graphs over finite fields and rings. Moreover, we will discuss some properties which our graphs have.

SUMMARY: We replace the problem of the RAID system in computer science with the problem of cyclic orderings in graph theory. We pay attention to constructions of cyclic orderings called cluttered orderings. There have been several studies on cluttered orderings of the complete bipartite graph \(K_{\ell ,\ell }\). Mueller et al. presented cluttered orderings in the case of \(\ell = 3t, 10t\). In this paper, we investigate cluttered orderings of the complete bipartite graph \(K_{\ell , m}\).

SUMMARY: We give a method for counting perfect matchings in line graphs. Consequently, for graphs \(G\) of maiximum degree at most 3, we give a closed formula for the number of perfect matchings in \(L(G)\).

SUMMARY: We shall determine the impurity of projective planar graphs. The concept of impurity is related to edge-maximal graphs.

SUMMARY: This talk introduces the integration theory with respect to the Euler characteristics of posets (categories), as a discrete analog of Baryshnikov and Ghrist’s work on topological Euler calculus. As its application, we consider the counting problem in sensor network theory. We enumerate targets which lie on an acyclic network graph with sensors detecting the targets, by using discrete Euler calculus.

SUMMARY: We have proposed the inclusion-exclusion integral which is an integral with respect to monotone measure and interaction operator. In this talk, we show a concrete way of construction of interaction operator based on t-norm and give a sufficient condition for monotonicity of the inclusion-exclusion integral. Moreover we give several examples of the operators which satisfy this sufficient condition.

SUMMARY: A lonesum matrix is a (0,1)-matrix which can be uniquely reconstructed from its row and column sums. We plan to talk about counting restricted lonesum matrices. We also discuss recurrence formulas for poly-Bernoulli numbers which derived from such counting formulas.

SUMMARY: A travel groupoid is an algebraic structure which has information of a graph and walks (paths) on the graph. We study a special travel groupoid called a simple non-confusing travel groupoid and characterize it by spanning trees on the graph associated to the simple non-confusing groupoid. We also count the number of simple non-confusing groupoids on cycle graphs.

SUMMARY: It was proved that any orthogonal polyhedron is continuously flattened by using a property of a rhombus. We investigated the method precisely, and found that there are infinitely many ways to flatten such polyhedra. In this talk, we prove that the infimum of the area of moving creases is zero for \(\alpha \)-trapezoidal polyhedra. As a by-product we provide a continuous flattening motion whose area of moving creases is arbitrarily small for more general types of polyhedra.

SUMMARY: An edge of a \(k\)-connected graph is said to be \(k\)-contractible if the contraction of the edge results in a \(k\)-connected graph. A \(k\)-connected graph with no \(k\)-contractible edge is said to be contraction-critically \(k\)-connected. Kawarabayashi showed that \(K_1+(P_3\cup K_2)\) is a forbidden subgraph of contraction-critically \(5\)-connected graphs. We present a new forbidden subgraph which has \(K_1+(P_3\cup K_2)\). This is an extension of the previous result due to Kawarabayashi.

SUMMARY: A pair \((G, c)\) of a graph and an edge-coloring \(c\colon E(G)\to \mbox {\textbf {N}}\) is called an edge-colored graph. If \(c\) is an injection, we say that \((G, c)\) is a rainbow graph. For a connected graph \(H\), an edge-colored graph \((G, c)\) is said to be rainbow \(H\)-free if \(G\) does not contain a rainbow subgraph which is isomorphic to \(H\). By definition, if \(H'\) is a connected subgraph of \(H\), every rainbow \(H'\)-free graph is rainbow \(H\)-free. In this talk, we report a reverse phenomenon. Let \(K_{1,k}^+\) denote the graph of order \(k+2\) which is obtained from \(K_{1,k}\) by performing a simple subdivision to one edge. Then we show that every rainbow \(K_{1,k}^+\)-free complete graph edge-colored in sufficiently many colors is rainbow \(K_{1,k}\)-free. We also show that for a connected graph \(H\) and a connected proper subgraph \(H'\), if every rainbow \(H\)-free complete graph edge-colored in sufficiently many colors is rainbow \(H\)-reee, then \((H', H)=(K_{1,k}, K_{1,k}^+)\) for some \(k\).

SUMMARY: In this talk, some recent results on degree conditions for properly colored cycles and rainbow cycles in edge-colored graphs will be reviewed.

SUMMARY: For a poset \(P\), the strict-double-bound graph (sDB-graph sDB(P)) is the graph on \(V(P)\) for which vertices \(u\) and \(v\) of sDB(P) are adjacent if and only if \(u\) is not \(v\) and there exist \(x\) and \(y\) in \(V(P)\) distinct from \(u\) and \(v\) such that \(x\) is a lower bound of \(u,v\) and \(y\) is an upper bound of \(u,v\). For a poset \(P\), \(P\) is a series parallel order if \(P\) contains no induced subposet isomorphic to the \(N\)-poset. We obtain the following result. For a series parallel order \(P\), if \(P_3\) is an induced subgraph of a component with at least four vertices of sDB(P), then \(P_3\) is contained \(C_4\), \(K_4-e\), \(K_{1,3}\) or \(3\)-pan as an induced subgraph.

SUMMARY: A tree-based network on a set X of leaves is said to be universal if any rooted binary phylogenetic tree on X can be its base tree. In my earlier work, the concept of universal tree-based networks was defined and it was shown that there exist infinitely many universal tree-based networks for any number |X| of leaves. In this talk, I will discuss the minimum size of universal tree-based networks. This talk is based on joint work with Satoru Fujishige (RIMS, Kyoto University) and Shizuo Kaji (Yamaguchi University, JST PRESTO).

SUMMARY: We define a quaternionic analogue of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we show a quaternionic analogue of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the manner of obtaining eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.

SUMMARY: We define an \((n+1)\)-variable Bartholdi zeta function and an \((n+1)\)-variable Bartholdi \(L\)-function of a graph \(G\), and give determinant expressions of them. We present a decomposition formula for the \((n+1)\)-variable Bartholdi zeta function of a regular covering of \(G\). Furthermore, we express the \((n+1)\)-variable Bartholdi zeta function of a regular covering of \(G\) as a product of its \((n+1)\)-variable Bartholdi \(L\)-functions.

SUMMARY: We introduce an extension model called partition-based quantum walk, which includes most quantum walk models driven by two local operators, such as the coined model, Szegedy’s model, and the 2-tessellable staggered model. We show that all those families of quantum walk models using two local operators are unitary equivalent. The new framework is based on two equivalence-class partitions of the computational basis, which establishes the notion of local dynamics.

SUMMARY: We propose a new version of quantum walks on simplicial complexes (named simplicial quantum walk), which is an alternative of preceding studies by authors. We show that it is unitary equivalent to a bipartite walk on associated bipartite graphs, coined quantum walk on a graph. Moreover, if simplicial complexes are orientable, the simplicial quantum walk is unitary equivalent to coined quantum walk on a graph with duplication structure.

SUMMARY: Here we show that the quantum search on the specific simplicial complex corresponding to the triangulation of \(n\)-dimensional unit square driven by this new simplicial quantum walk works well, namely, a marked simplex can be found with probability \(1+o(1)\) with in a time \(O(\sqrt {N})\), where \(N\) is the number of simplices with the dimension of marked simplex.

SUMMARY: A Bethe tree is a rooted tree such that in each level the vertices have equal degree. In this paper we focus on the periodicity of the Grover walk on Bethe trees. The Grover walk is a kind of quantum walks on graphs, and the time evolution operator of the Grover walk is determined by the graph. A periodicity is a special feature of quantum walk, and we have found some graphs to induce a periodic Grover walk. We find the classes of Bethe trees which induce a periodic Grover walk under an assumption.

SUMMARY: We consider a discrete-time 2-dimensional 4-state quantum walk. The evolution of the quantum walk is described by a unitary operator \(U\), which is the product of a space-dependent coin operator and a shift operator weighted with probabilities \(q_1\) and \(q_2\). Supposing that the coin operator and the shift operator are self-adjoint and unitary,

Fuda, Funakawa, and Suzuki proved that localization occurs \(|q| = \sqrt {|q_1|^2 + |q_2|^2}\) is sufficiently small. In this talk, we study about the localization without smallness of \(|q|\).

SUMMARY: In this talk, we consider a discrete-time one-dimensional two-state quantum walk called a split-step quantum walk. The evolution operator of the split-step quantum walk is defined by the product of a shift operator and a space-dependent coin operator. Weak limit theorem of the split-step quantum walk and its explicit limit distribution are presented.

SUMMARY: Wildberger gave a way to construct a finite hypergroup from a random walk on a certain kind of finite graphs. His method is applicable to a random walk on a certain kind of infinite graphs. In this talk, we formulate his method and give some examples that produce hypergroups and that do not produce hypergroups.

SUMMARY: The convergences of the extended Newton method to the roots of different initial values cause catastrophe, chaos, and fractals.

SUMMARY: In this talk, I will show you methods of data analysis using persistent homology and machine learning. Persistent homology enables us to describe the shape of data quantitatively from the viewpoint of homology and it is useful to study heterogeneous geometric structures. Machine learning enables us to detect characteristic patterns from data. By the combination of persistence homology and machine learning, we can quantitatively and statistically find characteristic geometric pattern hidden behind the data. Persistence Image and linear machine learning models are used for our methods. This combination gives us a very intuitive visualization of the learned result. “Inverse Problem” techniques for persistence diagrams are also effectively used to visualize the learned result.

SUMMARY: A integer-type algorithm for accurately solving linear ODEs by means only of four arithmetic operations among integers had been proposed by the author. Some direct ‘decipherments’ of numerical results by this algorithm enable us to see what is the essence of the accuracy of this algorithm, because this algorithm uses ‘exact’ Gaussian-integer-valued expansion coefficient sequences. By the decipherments, in this study, it turned out that this algorithm is closely related to number theory, in that a numerical expansion coefficient sequence by this algorithm is a rational linear combination of rational solution sequences of homogeneous linear difference equations with non-constant rational coefficients which accurately approximates their finite-norm irrational linear combination described in terms of algebraic extension of rational field. The relationship of this algorithm to continued fractions can be explained in this context.

SUMMARY: A computational method for equilibria of a nonlinear autonomous system is considered by using bisection method. The method led to analyze dynamical behavior and qualitative properties of the autonomous system.

SUMMARY: In this talk, we proposed a numerical method for estimating blow-up rate of blow-up solutions for a class of nonlinear evolution equations which have a scaling invariance. To use this scaling invariance we adopt the rescaling algorithm to the problems and numerically estimate the blow-up rates. Applying the method to several examples, we examine the effectiveness of the method.

SUMMARY: We propose a new shape derivative formula for contour integrals with logarithmic kernels which yields a numerical scheme to compute vortex patch equilibria. Owing to its simplicity, any steady configuration of point vortices can be extended to that of vortex patches. As a test problem, a periodic array of vortex patches is considered to show the efficiency of the new formula.

SUMMARY: The numerical analysis of the Cauchy problem for semi-linear Klein–Gordon equation in the de Sitter spacetime is considered. The solution of the equation expresses the property of expansion or contraction depending on conditions. Some of the terms in the equation present the dissipative and antidissipative effects. Since it is difficult to study the property and the effects analytically, we investigate them with numerical simulations. In addition, we study the numerical stability of the solutions.

SUMMARY: When we search for numerical solutions to the Einstein’s equation, we typically monitor the conservation of the constraints as a sanity check against numerical errors in the solution. The conservation of constraints is a necessary but not sufficient condition for the solutions to the Einstein’s equation. We propose a method by using the geodesic equation to improve the reliability of the solution.

SUMMARY: We propose a computer-assisted procedure to prove the invertibility of a linear operator in a Hilbert space and to compute a verified norm bound of its inverse. A number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates. The approach of the present talk is expected to bridge the gap between the two previous procedures in actual numerical verifications. Several verification examples that confirm the actual effectiveness of the proposed procedure are reported.

SUMMARY: We consider the finite element approximation for the linear homogeneous parabolic problem with the homogeneous Neumann boundary condition. We assume that the elliptic operator in the equation does not have lower order terms, that is, the operator is not positive definite. In this case, both exact and approximate solutions are globally bounded; thus the error is also bounded uniformly in time. However, in many literature, they use the Gronwall inequality, which causes the exponentially increasing term with respect to the time variable. As far as we know, there are no literature on time-global error estimates for these problems. In this talk, we present the time-global \(L^\infty \)-\(L^p\)-error estimates for sufficiently smooth initial data.

SUMMARY: It is useful to make numerical method for nonlinear PDEs in higher dimension for researching critical phenomena of it. So, we consider a spherically symmetric Poisson equation in \(N\)-dimensional ball. The previous study proposed finite element method using weight function \(x^{N-1}\) and showed optimal weighted \(L^2\) error estimate. However, there is a disadvantage of increasing error near origin. The another approach to use weight function \(x\) showed optimal \(L^\infty \) error estimate. In this paper, we see the PDE as singularly perturbed convection-diffusion equation through later approach, and apply discontinuous Galerkin method to it. We show some estimates and offer some numerical results.

SUMMARY: “Re”infection of recovered individuals, as a consequence of waning immunity and change of his/her susceptibility add further complexity in understanding disease transmission dynamics, forming a delayed feedback from infective population to susceptible population. In this talk we discuss dynamical aspects of a series of epidemic models, paying attention to reinfection dynamics, formulated by delay differential equations (DDE) and renewal equations (RE). We introduce a mathematical model by delay differential equations to provide a possible explanation of periodic outbreak of a childhood disease observed in Japan. Simple threshold dynamics is shown for a general SIS epidemic model formulated by a nonlinear renewal equation. On the other hand, we shall show that heterogeneous susceptibility can induce epidemic, after approaching to the trivial equilibrium.

SUMMARY: We study the stability and the bifurcation structure of standing pulse solutions to a singularly perturbed three-component reaction-diffusion system. We can show the detailed information of dependence on the parameters about the stability properties and the bifurcation structure.

SUMMARY: On a plane and a sphere, an \(N\)-ring is unstable for \(N>7\). We introduce on the inner side of a toroidal surface, however, an \(N\)-ring is stable when the aspect ratio of the torus is sufficiently large for any fixed \(N\).

SUMMARY: We prove the non-integrability of the spacial \(n\)-body problem. In order to prove it, we focus on the singularity of the extended differential equations and then apply the Morales–Ramis theory to it. We also discuss the non-integrability of the spacial restricted \(n+1\)-body problem.

SUMMARY: This talk is concerned with the Neumann problem of a diffusive Lotka–Volterra prey-predator system with finitely many protection zones for the prey species. We discuss the stability of non-negative constant solutions. Moreover, we study the existence and non-existence of positive stationary solutions by applying the bifurcation theory.

SUMMARY: We analyze the stationary localized patterns in a singularly perturbed three-component FitzHugh–Nagumo model. We derive explicit conditions for the existence and stability of these type of localized solutions by combining geometric singular perturbation techniques and an action functional approach. The action functional replaces the Melnikov integral approach and Evans function computation to derive existence conditions and critical information on the stability of the localized patterns.

SUMMARY: We study break down of bone metabolism, using mathematical modeling. The principal part of this model is composed of two pathways of maturation, that is, from pre-osteoblast to osteoblast and from pre-osteoclast to osteoclast. There is also a pathway of acceleration to the formation of pre-osteoclast by pre-osteoblast. This pathway is evoked by a cytokine, called RANKL. Experimental data, on the other hand, suggest a differentiation annihilation factor, to the maturation pathways above. Here we formulate the above feedback loops as a system of ordinary differential equations, pick up dynamical equilibria, and study their break down.

SUMMARY: The transcription factor NF-kB induces expression of multiple genes by shuttling between cytoplasm and nucleus. Previous studies have reported that the transcription of target gene is activated by phosphorylation of NF-kB, and decreased by dephosphorylation. In this study, we constructed a new mathematical model considering phosphorylation and mathematically analyzed how the phosphorylation of NF-kB effects on the oscillation phenomena. As a result, our new model, explained an appearance of a stable periodic orbit, which appeared in a transitional manner in response to the attenuation of an external stimulus, and also indicated that the NF-kB oscillation occurred by attracting to the periodic orbit.

SUMMARY: Annealing of copolymers has become a tool of great importance to reconfigure nanoparticles. We present experimental results of annealing copolymer nanoparticles and a mathematical model to describe the morphological transformation from lamellae to onion. A good correspondence between experimental findings and predictions of the model is observed. The model based on an appropriate free energy leads to a set of Cahn–Hilliard equations that correctly describes the dynamical transformation from lamellae particles to onion and reverse onion-like particles, regardless of the nature of the annealing process. This universality makes possible to describe a variety of experimental conditions involving nanoparticles underlying a heating process. A notable advantage of the proposed approach is that it makes possible to selectively control the interaction between the confined copolymer and the surrounding media.