2018年度年会(於:東京大学)

SUMMARY: We consider good configurations from the point of view of the theory of distance sets. A subset \(X\) of a Euclidean space is called a \(k\)-distance set if there exists exactly \(k\) values of distances between two distinct points in \(X\). The study of distance sets was initiated by Erdös (1946). One of the major problems in the theory of \(k\)-distance sets in the \(d\)-dimensional Euclidean space is to determine the largest possible cardinality \(g_d(k)\) of \(k\)-distance sets and classify the distance sets \(X\) satisfying \(|X|=g_d(k)\). Such \(k\)-distance sets are said to be optimal. Furthermore, we are also interested in characterization of \(k\)-distance sets with large points relative to \(k\). In this talk, we introduce some results for optimal \(k\)-distance sets and extremal problems on distance sets.

SUMMARY: We introduce the energy method for structure-preserving finite difference schemes which inherit the physical structures such as energy conservation or dissipative laws. Another aim is to give some useful properties for difference quotient which is compatible with the structure-preserving finite difference schemes. The method and properties enable us not only to take the problem with more general nonlinearity but also to improve proofs of error estimate between the numerical and exact solutions. In this talk, after explaining our procedure by using a simple example, several our recent results are introduced.

SUMMARY: In this talk, we provide a methodology of verified computing for solutions to evolution equations (nonlinear heat equations, 1-dimensional advection equations with variable coefficients, and the complex Ginzburg–Landau equations). Our methodology is based on semigroup theory, which is widely used in analytical studies and originated from pioneering works by Hille and Yosida. The main contribution of this study is to combine a “classical analysis” with “computer-assisted methods” to provide a numerical method of enclosing a solution for evolution equations. The combination of quantitative estimates arising from verified numerical computations and qualitative results obtained by classical analysis is expected to open the access to many unsolved problems by purely analytical means.

SUMMARY: A cluttered ordering is a kind of cyclic orderings, and can be used to minimize the number of disk operations in RAID system. Mueller et al. (2005) decomposed the complete bipartite graph into isomorphic copies of the special bipartite graph \(H(h; t)\), where \(h\) and \(t\) are positive integers. The special bipartite graph \(H(h; t)\) has each \(h(t+1)\) vertices as upper vertex set and lower vertex set. In this talk, we define the special bipartite graph \(H(h, k; t)\), where \(h\), \(k(h \neq k)\) and \(t\) are positive integers. The special bipartite graph \(H(h, k; t)\) has \(h(t + 1)\) vertices as upper vertex set and \(k(t + 1)\) vertices as lower vertex set. We present some edge labeling of the infinite families of \(H(1, 2; t)\).

SUMMARY: In this work we construct an infinite family, parametrized by dimension order \(d\), of non-isomorphic indecomposable persistence modules over the commutative ladder of length 5. We provide a family of bifiltrations of topological spaces whose \(H_1\) persistent homologies is the infinite family of persistence modules. Moreover, we provide Vietoris–Rips constructions of the family of bifiltrations. Our construction provides an elementary proof of the fact that the commutative ladders with length greater than or equal to 5 is representation-infinite. Furthermore, we aim by this example to illustrate that indecomposable persistence modules of high dimension as a representation may encode some interesting and easy to visualize phenomenon.

SUMMARY: In this talk, we give two inequalities concerning increasing families of finite graphs. Those inequalities are derived from the theory of quasi-orthogonal integrals developed by de Branges–Rovnyak and Vasyunin–Nikol’skii.

SUMMARY: There are some well known properties which random graphs satisfy with probability \(1\). Here we deal with two such typical properties of random graphs, namely, the \(n\)-existentially closed (\(n\)-e.c.) property and the pseudo-random property. Cameron and Stark remarked that the pseudo-random property does not necessarily imply the \(n\)-e.c. property for large \(n\). However there seems no results about the inverse relationship.
In this talk, we give the first construction of \(n\)-e.c. graphs without the pseudo-random property, which means that the \(n\)-e.c. property also does not necessarily imply the pseudo-random property for every \(n\).

SUMMARY: We study the homological properties of random simplicial complexes, which have received a lot of attention over the past several years. In particular, we obtain asymptotic behavior of lifetime sums of persistent homology for a class of increasing random simplicial complexes, which is a higher-dimensional counterpart of Frieze’s zeta function theorem for the Erdős–Rényi graph process. Main results include solutions to the questions on the Linial–Meshulam complex process and the clique complex process that were posed in the preceding study by Hiraoka and Shirai. One of the key ingredients of the arguments is a new upper bound of Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links, which is regarded as a quantitative version of the cohomology vanishing theorem.

SUMMARY: For a positive integer \(d\), the \(d\)-dimensional Chebyshev–Frolov lattice is the \(\mathbb {Z}\)-lattice in \(\mathbb {R}^d\) generated by the Vandermonde matrix associated to the roots of the \(d\)-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev–Frolov lattices in axis-parallel boxes when \(d = 2^n\) for a non-negative integer \(n\), since the points are used for the nodes of Frolov’s cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. The existing enumeration algorithm for such points by Kacwin, Oettershagen and Ullrich is efficient up to dimension \(d=16\). In this paper we suggest a new enumeration algorithm of such points for \(d=2^n\), efficient up to \(d=32\).

SUMMARY: A continuous lattice is a semantic domain of a computation such as a lambda calculous. Our motivation comes from a development of a formal theory of semantic domains. We first talk about a theory of relational T-algebra, an extension of a T-algebra. Michael Barr proved the category of relational T-algebra defined by the ultra-filter monad is isomorphic to the category of topological spaces. We review their theory and reformulate it using our simple flamework of a relational calculus.

SUMMARY: We show that the Smith normal form of a skew-symmetric D-optimal design of order \(n\equiv 2\pmod {4}\) is determined by its order. We apply our result to show that certain D-optimal designs of order \(n\equiv 2\pmod {4}\) are not equivalent to any skew-symmetric D-optimal design.

SUMMARY: Let \(v(k,\lambda )\) be the maximum order of connected bipartite \(k\)-regular graphs whose second-largest eigenvalues are at most \(\lambda \). We show an upper bound for \(v(k,\lambda )\), which is based on the linear programming bound. If a graph attains the bound, then it is a distance-regular graph that satisfies \(g\geq 2d-2\), where \(g\) is the girth and \(d\) is the diameter of the graph. There are infinitely many bipartite distance-regular graphs that satisfy \(g\geq 2d-2\). We can prove the non-existence of bipartite distance-regular graphs with \(g\geq 2d-2\) for \(d>26\) by the manner of Fuglister (1987). This is a joint work with Sebastian Cioabă and Jack Koolen.

SUMMARY: A subdigraph \(H\) of an arc-colored digraph \(D\) is properly colored if any two consecutive arcs of \(H\) receive distinct colors. A kernel by properly colored paths of an arc-colored digraph \(D\) is a set \(S\) of vertices of \(D\) such that (i) no two vertices of \(S\) are connected by a properly colored directed path in \(D\), and (ii) every vertex outside \(S\) can reach \(S\) by a properly colored directed path in \(D\). We conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively.

SUMMARY: A matching \(M\) of a graph \(G\) is said to be extendable if \(M\) is a subset of a perfect matching of \(G\), and \(M\) is said to be distance \(d\) matching if the edges of \(M\) lie pair-wise distance at least \(d\). If every distance \(d\) matching of \(G\) is extendable, then we say that \(G\) is distance \(d\) matchable. In this talk we introduce the following results: 1) Let \(G\) be a \(3\)-connected cubic bipartite graph. If there exist two cycles \(C_1\), \(C_2\) of length at most \(d\) such that \(E(C_1) \cap E(C_2) = \{e\}\) for every \(e \in E(G)\), then \(G\) is distance \(d\) matchable. 2) Let \(G\) be a \(3\)-connected cubic bipartite planar graph. If there exist two cycles \(C_1\), \(C_2\) of length at most \(6\) such that \(e \in E(C_i)\) (\(i=1,2\)) for every \(e \in E(G)\), then \(G\) is distance \(6\) matchable.

SUMMARY: A triangulation on a closed surface is a graph embedded on the surface each of whose face is triangular. 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 numbers \(n\) such that \(G\) has \(n\)-triad colorings. The set can be determined completely by the chromatic number of \(G\) if \(G\) is embedded on the sphere or the projective plane. In this talk, we shall focus on \(G\) which is embedded on the torus and investigate the above set of \(G\).

SUMMARY: An r-dynamic k-coloring of a graph \(G\) is a proper \(k\)-coloring such that any vertex \(v\) has at least min\(\{r,\deg _G(v)\}\) distinct colors in \(N_G(v)\). The r-dynamic chromatic number \(\chi _r^d(G)\) of a graph \(G\) is the least \(k\) such that there exists an \(r\)-dynamic \(k\)-coloring of \(G\). Loeb and et al, proved that \(\chi ^d_3\leq 10\) if \(G\) is a planar graph, however this result is not considered not to be sharp. Thus finding an optimal upper bound on \(\chi ^d_3(G)\) for a planar graph \(G\) is a natural interesting problem. We will show some upper bounds on \(\chi ^d_3(G)\) for triangulations on the plane, the projective plane and the torus.

SUMMARY: It is well-known that every 3-connected planar graph is uniquely embeddable on the sphere but it is not uniquely embeddable on any surface other than the sphere. We shall focus on a 3-connected 3-regular planar graph and classify structures of its embeddings on the torus, the projective plane and the Klein bottle.

SUMMARY: Let \(G\) be a \(k\)-regular \(k\)-edge colorable graph. Moreover, let \(\varphi \) be a \(k\)-edge coloring of \(G\). Let \(v\) be a vertex of \(V(G)\) and \(E(v)\) denotes the set of all edges incident with \(v\). Let \(\rho _v\) be a bijective map \(E(v) \to \{1,2,...,k\}\) and we call the set \(\rho = \{\rho _v: v \in V(G)\}\) basis. Now we obtain the correspondence \(\pi _v = \varphi \circ \rho _v^{-1}\) for every \(v\). The signature of \(\varphi \) is defined as \(\prod _{v \in V(G)}\) sign\((\pi _v)\).

In this talk, we obtain the signatures of \(k\)-edge-colorings in \(k\)-regular graphs on the projective plane for a certain basis \(\rho \). Moreover, this has an application to the list coloring conjecture for certain graph class by using former result.

SUMMARY: We consider a spanning bipartite subgraph of an even (i.e. Eulerian) triangulation \(G\) on a surface. If \(G\) has a spanning bipartite quadrangulation, then its size is two-thirds of \(E(G)\). In this talk, we discuss the condition of \(G\) having a spanning bipartite quadrangulation \(Q\). We also discuss the maximum size of a spanning bipartite subgraph of \(G\) when \(G\) does not have \(Q\).

SUMMARY: In our recent research, we proved that two balanced (or 3-colorable) triangulations of a closed surface are not necessary connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. In this talk, we especially discuss two local operations called a pentagon contraction and a pentagon splitting, which are also defined for balanced triangulations of closed surfaces. We show that most two balanced triangulations of a closed surface are transformed into each other by a sequence of the above two operations. Furthermore, we introduce such exceptional balanced triangulations of closed surfaces with low genera.

SUMMARY: We define a weighted generalized Bartholdi zeta function and a weighted generalized Bartholdi \(L\)-function of a digraph, and present their determinant expressions. Furthermore, we give express the weighted generalized Bartholdi zeta function of a group covering of a digraph by a product of its weighted generalized Bartholdi \(L\)-functions.

SUMMARY: We consider the two-state space-inhomogeneous coined quantum walk (QW) in one dimension. We obtain the uniform measure as the stationary measure by solving the eigenvalue problem. This approach is based on the method giving by Kawai, Komatsu and Konno. (2017).

SUMMARY: We analyze two types of the Quantum walk on the 2-dimensional torus, the Grover walk and the Fourier walk, and obtain provability amplitude and the search algorithm using the Fourier walk.

SUMMARY: We consider the 2-dimensional 4-states quantum walk. This quantum walk is an extension of the 1-dimensional split-step quantum walk. By the earlier study, it is known that if a particular function \(f\) has zero points, then time evolution operator \(U\) has eigenvalues. In such a case, localization occurs. In this talk, we introduce the necessary and sufficient condition of \(f\) has zero points. This research is a joint work with T. Fuda, S. Sasayama and A. Suzuki.

SUMMARY: We consider the property of lazy Fourier walk in one dimension. For example, stationary measure, time-averaged limit measure, and periodicity.

SUMMARY: This study investigates the unitary equivalence classes of one-dimensional quantum walks. We determine the unitary equivalence classes of one-dimensional quantum walks, two-phase quantum walks with one defect, complete two-phase quantum walks, one-dimensional quantum walks with one defect and translation-invariant quantum walks.

SUMMARY: Stationary measures of quantum walks on the one-dimensional integer lattice are well studied. However, the stationary measure for the higher dimensional case has not been clarified. In this talk, we give the stationary amplitude for quantum walks on the higher-dimensional integer lattice with a finite support by solving the corresponding eigenvalue problem. As a corollary, we can obtain the stationary measures of the Grover walks.

SUMMARY: We consider the two-state space-inhomogeneous coined quantum walk in one dimension. For a general setting, we obtain the stationary measures of the quantum walks by solving the eigenvalue problem. As a corollary, stationary measures of the multi-defect model and space-homogeneous quantum walks are derived.

SUMMARY: The Grover walk is a kind of quantum walks on graphs, and it is applied to various study fields. The Grover walk is determined by a unitary time evolution operator given by the underlying graph. So we can say that the Grover walk is induced by the graph. We focus on characterizations of graphs inducing periodic Grover walks, that is, there exists a integer \(k\) such that \(k\)-th iteration of the time evolution operator becomes identity operator. In previous walk, such graphs have been found, e.g. cycle graphs, path graphs, complete bipartite graphs. In this talk, we construct new graphs with these graphs, and consider the periodicity of the Grover walk on these graphs.

SUMMARY: In topological data analysis, the topological structures in data are transformed into a persistence diagram, and its statistical method is proposed by the Persistence weighted Gaussian kernel (PWGK). Here, let us consider the expectation of persistence diagrams by the PWGK. While it is difficult to calculate the expectation in general, the confidence set which contains the true expectation can be constructed by the bootstrap method. In this talk, we will discuss the bootstrap method by the PWGK and its asymptotic consistency.

SUMMARY: The theory of homology induced maps of correspondences proposed by Shaun Harker et al. is a powerful tool which allows the retrieval of underlying homological information from sampling data with noise or defects. In this study, we redefine induced maps of correspondences within the framework of quiver representations, and provide more concise proofs of the main theorems in the original paper. With this point of view, we easily extend these ideas to filtration analysis, which provides a new method for analyzing dynamical systems.

SUMMARY: Persistent homology (PH) is a significant tool for topological data analysis, which analyzes shape of data efficiently and quantitatively. A persistence diagram (PD) is a visualization tool of PH, which is a multiset on \(\mathbb {R} \times (\mathbb {R}\cup \infty )\). Each point on a PD (called a birth-death pair) corresponds a homological structure such as a ring, cavity, etc. appearing in the data. For a practical application of PH, we want to identify such a homological structure for a selected birth-death pair. The idea of a volume optimal cycle and the computation algorithm are proposed in this talk.

SUMMARY: We present a methodlogy using computation homology for quantitatve measurements in medical science, in collaboration with diagnostic doctors. Our talk will consits of two folds: cubical homology index for bone morphometry in three-dimension, and immunohistochemical scoring based on persistent homology.

SUMMARY: The Conley index over a base [Mrozek Reineck, Srzednicki 1997] is a generalization of the Conley index for flows [Conley 1972]. In the case when the base is the circle, it is naturally related to the discrete type of Conley index of a Poincare section of the flow. We will consider the homology version of the Conley index over a base, and discuss the relation between them.

SUMMARY: We extend the complex Newton’s method. We give the followings for the extended complex Newton’s method. Relationship between extended complex Newton’s method and Riemann surface. Conditional expression of initial values for convergence of extended complex Newton’s method. Speeds of convergences of extended complex Newton’s method. The distributions of roots of extended complex Newton’s Method.

SUMMARY: An integer-type algorithm for solving ODEs was proposed by the author and M. Hayashi. This algorithm is based on the expansion of solution functions by rational-function-type basis functions, and it is based on the ‘exact’ kernel vectors of non-square matrices. In this algorithm, we can read and ‘decipher’ integer coefficient sequences directly, and hence we can analyze behavior of numerical solutions exactly and pure-mathematically. In this study, the author gives further numerical examples where it is clearly shown that many hyperfunction components are contained in extra solutions mixed in numerical solutions obtained by this algorithm, by a direct ‘decipherment’ of integer coefficient sequences contained in numerical results by this algorithm, than the examples already presented.

SUMMARY: In this talk, we consider unbiased simulation methods functionals of solutions of one-dimensional reflected stochastic differential equations. Alfonsi–Hayashi–Kohatsu propose an unbiased simulation method for the present problem based on the parametrix method. However, the variance of this method is not finite in general unless one uses an importance sampling method. We propose a different way of obtaining what we call a second order parametrix method which leads to an alternative unbiased random variable with finite moments. We call this method “Second order unbiased simulation method for reflected stochastic differential equations”.

SUMMARY: This talk presents an computer-assisted procedure for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. This approach is an improvement of a theorem (Nakao, et al. 2015, Jpn. J. Ind. Appl. Math. 32, 19–32) that uses projection and constructive a priori error estimates. Several examples which confirm the actual effectiveness of the procedure are reported on.

SUMMARY: We study the existence of unimodal stationary solutions to the Proudman–Johnson equation with an external force. In particular, we are interested in the case of a high Reynolds number. In order to prove the existence and unimodality of a solution, we resort to interval arithmetic. We formulate the stationary problem for the Proudman–Johnson equation as a system of first order ordinary differential equations, and we apply the shooting method and the interval Newton method for proving the existence of a solution. As the shooting method is numerically unstable, we encounter some difficulties especially when the Reynolds number is high. For solving this problem, we apply the multiple shooting method and the multiple-precision floating-point arithmetic.

SUMMARY: We study the application of the Nitsche’s method to the parabolic problems. Under some assumptions, the parabolic initial-boundary value problem has a unique weak solution. The problem is discretized in space by the Galerkin method and the Dirichlet boundary condition is enforced weakly by the Nitsche’s method. It is well known that the bilinear form satisfies the Galerkin orthogonality. In this presentation, we will prove that the bilinear form also satisfies the inf-sup condition. This condition implies that the resulting semi-discretized problem has a unique solution. Moreover, the error estimate follows directly from the inf-sup condition and the Galerkin orthogonality.

SUMMARY: We introduce a numerical method for biharmonic problems which is obtained by applying the interior penalty method to a mixed nonconforming finite element method which is called the Hermann–Johnson (HJ) method. We show that a priori error estimates of our method can be the same as those of the HJ method by appropriately choosing a penalty parameter in our method.

SUMMARY: For the Stokes equation defined in 3D domain with a general shape, the Scott–Vogelius finite elements are used to obtain strictly divergence-free approximation solution. Then, by using the hypercircle equation method, a quantitative a priori error estimation is obtained for the FEM solution. Such an a priori error estimation can be used in solution existence verification of nolinear Navier–Stokes equation defined in 3D domain with general shapes. The convergence rate is confirmed by numerical results.

SUMMARY: The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition for the DG bilinear form. Then, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new. It differs from previous works on the DG time-stepping method by which the method is formulated as the one-step method.

SUMMARY: The discontinuous Galerkin time-stepping method (DG time-stepping method) is a time-discretization method based on the discontinuous Galerkin finite element method. In contrast to one-step methods, the approximated solution is well-defined at each time in the DG time-stepping method. Therefore, it gives an efficient numerical algorithm with space-time methods for moving boundary problems such as fluid structure interaction. However, there are few studies on theoretical analysis for the behavior of approximated solutions at each time. In this talk, we address the DG time-stepping method for parabolic problems in the framework of analytic semigroup theory. We present optimal order error estimates for the homogeneous heat equation. The key point is rigorous estimates for rational functions that express the approximated solutions.

SUMMARY: We previously reported examples of convergence of unstable finite difference schemes applied to quasilinear partial differential equations of the normal form with analytic initial data. This time we report that convergence also holds for initial value problems for the heat equation with initial data in a certain class of analytic functions and also for the sideways heat equation with initial data in a Gevrey class.

SUMMARY: We investigate the numerical stability for semi-linear Klein–Gordon equations in de Sitter spacetime. We show the differences of the numerical stability between some positive nonlinear terms and the negative one. In addition, the numerical stability in the four dimension and more is shown.

SUMMARY: The energy-preserving method based on the variational principle and the discrete gradient method are methods for designing a scheme for the Hamilton equations that preserves the energy conservation law exactly. In the discrete gradient method, the various extensions, e.g. the extension to the equations on a manifold have been proposed. Although the method based on the variational principle has an advantage in the view of the computational cost, this method has not been applied to the equations on the manifold yet. In this talk, we extend the energy-preserving method based on the variational principle to equations on the Lie group and show a numerical test for the heavy top problem.

SUMMARY: We propose nonlinear and linear finite difference scheme for the conservative non-local Allen–Cahn equation. Both proposed schemes inherit characteristic properties, the conservation of mass and the decrease of the global energy from the equation. We show that the schemes are stable in the sense that the numerical solution is bounded concerning max-norm, and have a unique solution. Since the nonlinear scheme is the system of equations concerning the new time step, it takes time to compute. Numerical examples demonstrate the effectiveness of the proposed scheme and that the computational time of the linear scheme is shorter than one of the nonlinear scheme. In this talk, we mainly introduce the linear scheme.

SUMMARY: Free boundary problems are recently used to model phenomena of biological invasion for species such as migration into a new habitat (e.g., Du & Lin (2010) and references therein). These ideas are also applied to epidemic models. In this talk, we extend the result in Kaellen (2017) to the simple diffusive epidemic model with free boundary, namely we prove the existence of a semi wave solution. We numerically observe the semi wave and the front motion of this model with free boundary.

SUMMARY: Many researchers have studied the self-driven particles. In one example, there is camphor atop water channel. It is now said that the motion of camphor is caused by differences in surface tension. The gradient of surface tension is induced by a camphor molecular layer development atop the surface. Mathematical models for the camphor motion have been constructed used the above mechanisms, and the models reproduce many characteristic motion. Although Marangoni Convection seems to influence the self-motion of the camphor, there are only a few reports discussing mathematical models that include convection explicitly. We have constructed a mathematical model for the self-motion of camphor including influence of convection, and now are calculating some cases to compare with some experiments.

SUMMARY: We propose a mechanism of stable formation of the granular layer in the epidermis. In our mathematical model of the epidermis, we assume that a stimulant, which promotes the differentiation process of epidermal cells, is released when cells undergo cornification. We demonstrate that our model forms the granular layer and confirm that its layer structure is maintained stably by using some cost functions. We are also working on formation of tight junctions, which exit in the granular layer and play an important role in skin barrier function, and we introduce our trial models for them.

SUMMARY: Recently, it has been attended the relation between the elongation phenomena and rotational migration of cell group since they are observed in three-dimensional morphogenesis such as fruiting body formation of Dictyostelium Discoideum and somite formation of zebra fish. In particular, although it is known that the somite is covered by basement membrane, it is not clear the reason why rotational migration occurs. In order to understand cellular mechanism for rotational migration of cell groups covered by basement membrane, we propose a mathematical model which consists of a self-propelled particle model representing cellular migration and a phase-field model representing basement membrane. Moreover, we will show phase diagram of parameters for migration modes and give a theoretical suggestion for biological experiments.

SUMMARY: In this talk, we deal with the initial value problem of evolutionary differential equations with a mixed derivative on the periodic domain. Here, “mixed derivative” indicates the case where a spatial differential operator is operating on the time derivative, obscuring the vector field describing the flow. Therefore, some reformulation to reveal it is the first step of PDE-theoretical and numerical studies. However, it is nontrivial because the spatial differential operator is not invertible and cannot be easily eliminated. Though this issue was already settled for linear cases, general theory has been undeveloped. In this talk, we propose a novel procedure for wider class of equations. Moreover, as an application, we establish the global well-posedness of the sine-Gordon equation in characteristic coordinates.

SUMMARY: Uniform flow is one of fundamental steady solutions of Euler equation on a plane. A generalization of the flow on curved surfaces is a Killing vector field, which is also a steady solutions of Euler equation on Riemannian manifolds. In this talk, we introduce how surfaces has a no-normal regular Killing vector field and as its application, construction of an analytic formula of Green’s function on the surface.

SUMMARY: We propose a one-dimensional hydrodynamic partial differential equation. This model is based on a Constantin–Lax–Majda–De Gregorio model generalized by Okamoto, Sakajo and Wunsch. The equation admits an inviscid invariant quantity. In the presence of the viscosity and a large-scale random forcing, the solution gives rise to a turbulent state with cascade of the inviscid invariant. We will give how those phenomenon is understood from the view point dynamical system.

SUMMARY: Based on the Linear-Quasi-Gaussian compensator, we design a linear feedback system stabilizing point vortex equilibria near an aerodynamic wing with two auxiliary flaps known as a Kasper Wing in the presence of a uniform flow. This is modeled by a two-dimensional incompressible an inviscid flow. The actuation mechanism is blowing and suction localized on the main plate represented as a sink- source singularity, whereas we measure pressure across the plate as system output. We show that the linearised system around these equilibria are both controllable and observable for almost all actuator and sensor locations. Numerical computations illustrate that Kasper Wing configurations are in general not only more controllable than their single plate counterparts, but also acquire larger basins of attraction owing to the feedback control.

SUMMARY: Generalized J-integral is the tool which is effective to study the shape optimization of singular points (containing boundary) with respect to various cost functions, energy, mean compliance, least square errors, in boundary value problems for partial differential equations. I will talk an application of Generalized J-integral method to shape sensitivity of eigenvalue problems.

SUMMARY: We study synchronization of two interacting populations of oscillators. We assume that the sign and the strength of the interactions are taken as system variables, and that the variables are determined by a function of the oscillator variables. Under an appropriate choice of the function, we can observe spontaneous intra- and inter-group phase synchronization.

SUMMARY: Three-component FitzHugh–Nagumo equation is investigated. This equation has a parameter region in which traveling pulse with oscillatory tail appears. When such moving pulse interacts with heterogeneity of the media, the interaction between tail and heterogeneity is important. We consider a bump-type heterogeneity and investigate the asymptotic behavior of the pulse motion when a pulse collides with the bump. When the width of the bump is fixed and the height of that is changed, three different asymptotic behaviors arise sequentially: Oscillatory pinning (OSC), stationary pinning (STA), and rebound (REB). In these asymptotic states, it is conjectured that all asymptotic states are contained in the set of bifurcation branches which is generated from the trivial branch.

SUMMARY: The dynamics of two self-propelled camphor disks on an annular field is explored both numerically and analytically. In our previous study, it was already found by the direct numerical simulation of a model equation that the two camphor disks exhibited a variety of behavior, and underwent transition between the behavior as the length of the annular field was varied. In order to analytically elucidate the mechanism for the transition in behavior, we reduced the model equation which consisted of two ODEs and one PDE into three ODEs for the motion of the two camphor disks. In this talk, the bifurcation structure that causes the transition will be revealed, based on the reduced ODEs.

SUMMARY: We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits. We show that if the sufficient conditions for real-meromorphic nonintegrability hold, then the stable and unstable manifolds of the periodic orbits intersect transversely, are quadratically tangent or do not intersect in general, and they do not intersect when the Hessian matrix of the Hamiltonian has a different number of positive eigenvalues at the associated saddle-centers. Our theory is illustrated for a system with quartic single-well potential.

SUMMARY: Here, we consider the derivative embedding of a scalar function. Since the embedded surface can hold some characteristics of the original time-series, one can derive some predictability result from characteristics of the embedded surface.

SUMMARY: We study a blow-up curve for the one dimensional wave equation \(\partial _t^2 u- \partial _x^2 u = 2^p|\partial _t u|^p\) with the Dirichlet boundary condition. The purpose of this talk is to show that the blow-up curve \(T\) satisfies that \(T'(x)\rightarrow -1\) as \(x\rightarrow 0 + 0\ (1)\) under the suitable initial conditions. To prove the result, we convert the equation into a first order system, and then present some numerical investigations of the blow-up curves. From the numerical results, we were able to confirm (1) holds numerically. Moreover, under some assumptions, we were also able to confirm (1) holds mathematically.

SUMMARY: In this talk, we consider properties of backward self similar solutions for a quasi-linear parabolic equation \(v_t=v^{\delta }(v_{xx}+v)\). Their properties are very important to investigate asymptotic behavior of solutions to this parabolic equation, especially, the blow-up sets and rates.

SUMMARY: Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates not only of so-called “type-I” but also other types.

SUMMARY: Geometric treatments of oscillatory blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. As in the preceding talk, we apply compactification of phase spaces and time-scale desingularization to characterization of blow-up solutions. In particular, when divergent solutions are characterized by trajectories on center manifolds of non-hyperbolic periodic orbits on the horizon for desingularized vector fields, they blow up in finite time with infinitely fast oscillation and faster blow-up rate than type-I rates in typical cases, while they can also grow up in infinite time in some cases. We see such behavior in a certain system known as Liénard equation.