2018年度秋季総合分科会(於:岡山大学)

SUMMARY: Quantum walk (QW) is a quantum version of random walk and has been extensively studied since around 2000. A striking property of QW is the spreading property. The standard deviation of the walker’s position grows linearly in time, quadratically faster than random walk, i.e., ballistic spreading. On the other hand, a walker stays at the starting position: localization occurs. Moreover, due to the rapid development of quantum computer by huge IT companies recently, it has become a reality that programs based on QWs run on quantum computers. Therefore, my title is “Quantum Walk 2.0”. In this talk, after an overview of the history of QW, I explain the recent trends.

SUMMARY: Despite the success of deep learning in practice, a lot of open problems remain in theory. For example, why deep learning succeeded to estimate a good neural network in spite of the large number of parameters, or what a neural network does in its “black-box” network. In this talk, I review recent developments on deep neural network theory; and explain how neural networks approximate functions from the viewpoint of the integral representation theory and ridgelet analysis; and what functions neural networks approximate from the viewpoint of the optimal transport theory and Wasserstein geometry.

SUMMARY: In this talk, we overview the recent developments of deep learning theory especially from the point of view of generalization and representation ability, and give some generalization error analysis from kernel methods and its applications to the model determination analysis. Along with rapid development of deep learning applications, its theoretical analysis has been developed extensively these days. In the first part, we overview the recent progress of deep learning theories. Second, we show what kind of quantity determines the generalization error. To do so, we define an intrinsic dimensionality from a kernel method perspective. Based on the quantity, we derive a generalization error bound. As an application of the theory, we show an approach for model structure determination, especially, model compression method. Finally, we analyze representation ability of deep learning using wavelet analyses.

SUMMARY: In this talk, we discuss the well-posedness of Cahn–Hilliard system with dynamic boundary condition. In this model, a characteristic property of conservation holds which is related to the sum of the volume in the bulk and on the boundary. By virtue of the effective usage of this property we can prove the existence and uniqueness of the solution and its continuous dependence. Moreover, we also treat a degenerate parabolic equations as asymptotic limits of Cahn–Hilliard systems.

SUMMARY: Persistent homology is a mathematical tool to characterize the shape of data quantitatively and efficiently. Persistence diagrams are used to visualize the information about persistence diagram and they are good descriptors of the shape of data. In the last decade, persistent homology is rapidly developed from theories and algorithms, to the applications to life scieces, sensor network, and materials science. From the viewpoint of mathematics, persistent homology is the homology theory on filtrations, and its generalization. However, to solve the practical problems, we require the power of computers and we need the ideas from computer science. In this presentation, I will talk about inverse analysis on a persistence diagram. The main topic is volume optimal cycles. The topic relates many mathematical and computational concepts, such as algebraic topology, linear programming, sparseness, etc. We also show the ability of inverse analysis combined with machine learning.

SUMMARY: For any general graphs, we provide a formula for the number of perfect matchings in line graphs.

SUMMARY: The topological complexity is a numerical invariant closely related to the robotic motion planning problem. I will present topological and combinatorial methods for calculating the topological complexity of a finite space and the associated simplicial complex.

SUMMARY: fMRI designs have been researched to be applied to investigate brain activity. In 2017, Lin, Phoa and Kao defined circulant almost orthogonal arrays (CAOAs) as a kind of fMRI designs. Recently, Satake, Yoshida and Sawa gave a systematic construction of CAOAs by using the idea of Hadamard 3-design. In their construction from Paley matrices, one must know the distribution of quadratic residues (QRs) modulo primes. The disribution of QRs has been attracted many mathematicians, however, many unknown parts are remained. In this talk, we give an observation of the distribution of QRs. From this result, a necessary conditon for existence of CAOAs, consructed from Paley matrices, can be obtained.

SUMMARY: We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose–Mesner algebra of commutative nonsymmetric association schemes. We give nonsymmetric association schemes of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose–Mesner algebra of its association schemes. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

SUMMARY: In 1984, Frankl and Füredi asked what is the maximum number of hyperedges in an \(r\)-uniform hypergraph with \(n\)-vertices such that every set of \(r+1\) vertices contains \(0\) or exactly \(2\) hyperedges. We consider this problem for the case that \(r=4\) and hypergraphs are obtained from tournaments.

SUMMARY: In this talk, we introduce the concept of the complete cycle index and discuss a relation with the complete weight enumerator in coding theory.

SUMMARY: We introduce the concept of the Tutte polynomials in genus \(g\) and discuss some properties. We note that the Tutte polynomials in genus one are well-known the Tutte polynomials. It is known that the Tutte polynomials are matroid invariants and we claim that the Tutte polynomials in genus g are also matroid invariants. The main result of this talk is to give inequivalent matroids which have same Tutte polynomial, and have different Tutte polynomials in genus \(2\).

SUMMARY: Let \(G\) be a graph. A mapping \(c: E(G) \rightarrow \mathbb N\) is called an edge-coloring of \(G\) and \(c(e)\) is called the color of an edge \(e\). A subgraph \(H\) of \(G\) is called properly colored, or shortly PC, if every pair of adjacent edges in \(H\) have distinct colors. Gallai gave a structure of edge-colored complete graphs which has no PC cycle of length three. For edge-colored complete bipartite graphs, Axenovich, Jiang and Tuza showed that if every vertex is incident with at least three edges whose colors are mutually distinct, then the complete bipartite graph contains a PC cycle \(C_4\) of length four. In this talk, we will discuss structures of edge-colored complete bipartite graphs which has no PC \(C_4\).

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. For a graph \(G\) and a vertex \(x\) of \(G\), let \(G[N(x)]\) be the subgraph induced by the neighborhood of \(x\). We prove that if \(G[N(x)]\) has less than \(\lceil \frac {k}{2} \rceil \) edges for any vertex \(x\) of a \(k\)-connected graph \(G\), then \(G\) has a \(k\)-contractible edge. We also show that the bound \(\lceil \frac {k}{2} \rceil \) is sharp.

SUMMARY: An \(n\)-coloring \(c : V(G) \to \{1, \ldots , n\}\) of a graph \(G\) on a surface is a facial \(t\)-complete \(n\)-coloring if every \(t\)-tuples of colors appears on the boundary of some face of \(G\). The maximum number \(n\) such that \(G\) has a facial \(t\)-complete \(n\)-coloring is called the facial \(t\)-achromatic number. They are expansion of a complete coloring and the achromatic number, respectively. In this talk, we will show some results of a facial \(3\)-complete coloring and the facial \(3\)-achromatic number of even triangulations on the sphere.

SUMMARY: A seminal theorem of Tutte states that planar 4-connected graphs are Hamiltonian. Applying a result of Thomas and Yu, one can show that every 4-connected graph with crossing number 1 is Hamiltonian. In this talk, we continue along this path and prove the titular statement. We also discuss the traceability 4-connected graphs with small crossing number.

SUMMARY: Let \(G\) be a (finite, simple, undirected) graph with vertex set \(V\). Let \(N(A; B)\) denote the set of all the vertices which are adjacent to every vertex in \(A\) but no vertex in \(B\) for disjoint \(A, B \subseteq V\). Let \(n\) be a positive integer. If \(N(A;B) \neq \emptyset \) for any pair of disjoint \(A, B \subseteq V\) (possibly empty) with \(|{A \cup B}| = n\), the graph \(G\) is said to be \(n\)-existentially closed (\(n\)-ec). Moreover, if \(N(A;B)\)’s never coincide for different choices of such \((A,B)\), then \(G\) is said to be \(n\)-separably existentially closed (\(n\)-sec). In this talk, I will introduce the relation between \(n\)-ec graphs, \(n\)-sec graphs, and some explicit constructions.

SUMMARY: In this talk, we treat a quantum walk model on a finite connected graph with infinite tails with supplying infinite energy. In this model, we can construct the stationary state of quantum walk corresponding to the stationary distribution of random walks and observe convergence to the state like random walks. We consider the stationary state for this model on some graphs with useful properties. In particular, we give the explicit expression of the stationary state of this model on hyper cubes and Hamming graphs, classes of distance-regular graphs, by using a property of the stationary state.

SUMMARY: We prove the inequalities among the spectral radius \(\rho (A)\) of a finite graph \(X\), the radius of convergence \(R\) of its Ihara zeta-function \(Z_X(u)\), and the average degree \(\bar {d_X}\) of \(X\). These inequalities are posed by A. Terras in her book (2011) as research problems. Relating to Terras’ inequalities, we propose a new conjecture between \(R\) and some Rayleigh quotients.

SUMMARY: We set a dynamical system on a finite and connected graph induced by the Grover walk. To this end, we add additional infinite length tails to the original graph and keep providing an external energy at each time step from them. Then we obtain i) a uniquely existence of a fixed point; ii) the scattering way from the temporal and spatial global view point in the long time limit. The total mass in the long time limit of the internal graph is evaluated by the number of edges.

SUMMARY: We consider discrete-time quantum walks on cycle graph and path graph with isospectral (but not need the same) coin operators. By using spectral analysis of the time evolution operators for the quantum walks, we have fundamental results for the two cases. For the cycle graph cases, we find a classification criterion of quantum walks from the point of the periodicity. On the other hands, for the path cases, we find a formula for the time-averaged distribution which consists of the information of the corresponding random walk and eigenvalues of the coin operators.

SUMMARY: We consider the 2-dimensional 4-states quantum walk. This quantum walk is an extension of the 1-dimensional split-step quantum walk. The time evolution operator \(U\) is defined by the multiplication of the shift operator \(S\) and the coin operator \(C\). In this talk, we introduce the condition of \(C\) for resonance of \(U\). Moreover, we show that the resonance vector belongs to \(L^{\infty }\) space.

SUMMARY: We define a time operator with respect to a unitary operator. This definition is similar to that of strong time operator. We establish a general existence theorem on such a time operator. We construct a time operator with respect to the unitary operator of the Hadamrd walk.

SUMMARY: The Grover walk is kind of quantum walks and it is defined by a graph. This is studied in many fields and has many applications. Also, there are recently studies on periodicity of Grover walk on many graphs. For example, Higuchi–Konnno–Segawa–Sato determine the periodic strongly regular graphs in 2017 and Kubota–Segawa–Taniguchi–Yoshie determine periodic graphs in the generalized Bethe trees in 2018. Periodic graphs are extremely rare and it is hard to find them. On the other hand, by through search by MAGMA, we found some new periodic graphs and consider families including them and determine periodic graphs of those. In this talk, we especially talk examples of periodic trees and introduce an infinite family including them. This talk is based on joint walk with Yusuke Yoshie (Tohoku university).

SUMMARY: Recently, quantum walks are intensively studied in quantum physics and quantum computing. The behavior of the quantum walk is quite different from that of classical random walk, e.g., ballistic spreading and localization. In this talk, we give the stationary measures by using transfer matrices induced by eigenvalue problem for quantum walks on the one-dimensional. For example, there are three classes of the stationary measures. First one is the set of the measures with exponential type. Second one is the set of the measures with quadratic polynomial type. Last one is the set of the measures with periodicity. Especially, we discuss the condition to have periodicity.

SUMMARY: It is known that we can continuously flatten the surface of a 3-dimensional cube onto any of its faces by moving creases to change the shapes of some faces successively, following Sabitov’s volume preserving theorem. Let \(C_n\) be an \(n\)-dimensional cube with \(n>3\), and \(S\) be the set of its \(2\)-dimensional faces, in other words, the \(2\)-dimensional skeleton of the square faces in \(C_n\). We show that \(S\) can be continuously attened onto any face F of \(S\) such that the faces of \(S\) that are parallel to F do not have any crease, that is, they are rigid during the motion.

SUMMARY: Let \(P\) be a polygon on the plane and we consider a geometric quadrangulation of \(P\), that is, a geometric plane graph with each finite face quadrilateral which is obtained from \(P\) by adding straight segments in the interior of \(P\). Introducing a new notion for \(P\), called the spirality, we give a sufficient condition for \(P\) to admit a geometric quadrangulation. Moreover, we give a condition for \(P\) such that any two geometric quadrangulations of \(P\) can be transformed into each other by flipping edges.

SUMMARY: A graph is called 1-planar if it can be drawn in the plane such that each edge has at most one crossing point against the other edges. We consider connectivity and Hamiltonicity of 1-planar graphs. On the relationship between planar graphs and Hamiltonicity, very famous Tutte’ theorem states that every 4-connencted planar graph \(G\) is Hamiltonian (i.e., \(G\) has a spanning cycle). In this talk, we shall state an analogous result for 4-connected maximal 1-planar graphs.

SUMMARY: We consider a new impartial game, called a feedback game, on Eulerian graphs. In this talk, we introduce some known results and our results about the time-complexity of the game and the game on toroidal grid graphs.

SUMMARY: A dominating set of a graph \(G = (V, E)\) is a subset \(D\) of \(V\) such that every vertex not in \(D\) adjacent to some vertex in \(D\). The maximum number of disjoint dominating sets in a dominating set partition of a graph \(G\) is called domatic number \(d(G)\) of \(G\). \(G\) is said to be domatically full graph if \(d(G) = \delta (G)+1\) where \(\delta (G)\) is the minimum degree of a vertex of \(G\). We give a constraction of domatically full graphs by using Cartesian products of graphs.

SUMMARY: Some recent results on safe set problems in vertex-weighted graphs will be reviewed.

SUMMARY: We investigate left eigenvalues of quaternionic quantum walks on finite graphs. We apply the second weighted zeta function of a graph to obtain some left eigenvalues of quaternionic Grover walks. A quaternionic version of Sato’s determinant formula for the second weighted zeta function plays a important role in our results.

SUMMARY: Quantum walks on the graph have extensively studied from various perspectives. Several results are known for cases where the Grover matrix is adopted as the coin operator. However, when adopting the Fourier matrix as the coin operator, there are almost no corresponding results. In this talk, we focus on the periodicity and present a necessary condition for quantum walks to have a finite period. As an application of this result, we showed that the quantum walks do not have any finite period for some specific graphs: complete graph, cycle graph with selfloops.

SUMMARY: The stationary measure of the 2-state quantum walk in one dimension has intensively investigated and given some examples by Konno and Takei (2015), Kawai, Komatsu and Konno (2017). In the quantum walk in one dimension, we discovered that the states of quantum walker at positions \(x-1\) and \(x+1\) can be completely described by the state at position \(x\). In other words, by taking the state at the origin for all times as the initial state, all the quantum states are obtained. In this talk, we introduce such a position-evolution process and present some results given by this process.

SUMMARY: Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we discuss about the property of the eigenvalues of the discriminant for the time evolution matrix of a 2-tessellable SQW on a graph, and present eigenvectors for some of its eigenvalues.

SUMMARY: We introduce several methods to define the self-inductance of a single loop as the regularization of divergent integrals which we obtain by applying Neumann (or Weber) formula for the mutual inductance of a pair of loops to the case when two loops are identical.

SUMMARY: In this talk, we give results about the spatio-temporal patterns generated by a nonlinear two-dimensional symmetrical elementary cellular automaton. By using pre-fractal sets we show the relation between the spatio-temporal pattern and a singular function. We also calculate the fractal dimension of the boundary of its limit set.

SUMMARY: The SKT cross-diffusion equation is proposed by N. Shigesada, K. Kawasaki and E. Teramoto in 1979 to investigate segregation phenomena of two competing species with each other in the same habitat area. Y. Lou and W.-M. Ni derived limiting equations to see whether this effect may give rise to a spatial segregation or not, and to clarify its mechanism. It has been thought that the number of solutions of a stationary limiting equation seems to be at most two. However, we have found several parameter values numerically for which there exist three solutions. In this talk, we show numerical overviews of existence, non-existence, multiplicity and stability of solutions to the stationary limiting equation.

SUMMARY: A pressure-Poisson problem is used in numerical schemes such as MAC (marker and cell), projection and particle methods for solving the Navier–Stokes equations. We introduce the Stokes problem and a corresponding pressure-Poisson problem, and compare between the two solutions using boundary data.

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 Nitsche’s method and DG method. In this study, we show the analysis and some numerical results of Nitsche’s method for Poisson equations with a Robin boundary condition in a smooth domain.

SUMMARY: This paper presents error analysis of the finite element method for computing spherically symmetric solutions of semilinear heat equations. In particular, we establish optimal order error estimates in a weighted \(L^2\) norm for the symmetric formation and in the \(L^\infty \) norm for the non-symmetric formulation.

SUMMARY: We provide a numerical validation methodology for the finite difference discretization of \(u_t = u_{xx} + e^u\) with \(0\)-Dirichlet boundary condition. The main ideas are compactification of phase spaces and time-scale desingularization. In the current case, treatment of numerical validations with exponential growth is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can validate blow-up solutions of differential equations with such exponential nonlinearity as in previous works.

SUMMARY: We consider second-order elliptic operators with Dirichlet boundary condition. This talk presents some improvements for our previous approaches for determining the existence and bounds of the inverse of the operators. Our proposed approaches are based on constructive \(L^2\)-norm estimates of Laplacian and applications of the authors’ previous result related to a posteriori estimates of inverse operators which use projection and a priori error estimations.

SUMMARY: We study the 2D filtered Euler equations, which are the regularized 2D Euler equations derived by a spatial filtering method, and focus on the vortex sheet solution that is a weak solution with locally finite kinetic energy and vorticity in the space of Radon measure. We show that vortex sheet solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the regularization parameter.

SUMMARY: We introduce a theoretical analysis for point vortex dynamics on minimal surfaces. First, we gives both a minimal surface spanned by a given boundary configuration and its uniformizing map to obtain a global isothermal coordinate. Second, we compute the evolution of point vortices on the coordinate. Finally, we discuss how the boundary configuration and shapes of the minimal surface affects the dynamics of point vortices.

SUMMARY: The method of fundamental solutions (MFS for short) is a meshfree numerical solver for linear homogeneous partial differential equations with constant coefficients, and it has been applied to various problems in the plane. In this talk, we extend MFS to a surface being conformally equivalent to a flat surface in the plane. Moreover, we apply it to point vortex dynamics on minimal surfaces and construct efficient numerical scheme based on MFS.

SUMMARY: The symmetry of coupled oscillator circuits consisting of four identical oscillators expressed by an odd function is studied. The equations derived by converting coordinate system are analyzed, because the symmetry of the circuit is a dihedral group representation provided that the coupling characteristics are expressed by an odd function. From the symmetry of the equations derived by converting coordinate system, it is shown that there exist the origin whose eight values of the coordinate are zero, and there exist two types of equilibrium points: one is the equilibrium point whose six values of the coordinate are zero, and the other is the equilibrium point whose four values of the coordinate are zero. An analysis is performed for these equilibrium points.

SUMMARY: Several years ago, we proposed an integer-type high-accuracy method for calculating the eigenvalues of \(M\)-th order differential operators. This methods utilizes the discontinuity of the \(M-1\)-th order derivatives of the solutions of characteristic functions, which is allowed under the multiplication by a factor from the left. In this study, an improvement of this method is proposed, where we eliminate the true eigenfunction components from the solutions of characteristic equations and we calculate the eigenvalues by analyzing only the error components caused by the singularities. Moreover, we give some numerical examples which show that we are successful in reducing to a large extent the computational quantity.

SUMMARY: We show that a nonlinear delay differential equation, which is a Hutchinson–Wright equation with distributed delay, has a periodic solution of period two when the steady state is unstable. To find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke (1974). The periodic solution is expressed in terms of the Jacobi elliptic functions. An implication for a related disease transmission dynamics model is discussed.

SUMMARY: We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some nondegenerate condition, no matter how small the random forcing terms are. This result is very contrasting to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. We illustrate our theory for the Duffing oscillator subjected to the Ornstein–Uhlenbeck process parametrically.

SUMMARY: We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using Kovacic’s algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable. The obtained results are very general and reveal their rational nonintegarbility for the wide class of parametrically forced nonlinear oscillators.

SUMMARY: Dynamical reconstruction by using delay embedding is studied. This embedding method can be used for direct inference of a dynamical system by using time-series only. This reconstruction procedure can be interpreted as derivation of a single equation with respect to the particular embedded variable. Focusing on polynomial dynamical systems, Gröbner basis plays an important role to characterise this methodology. Corresponding members of the resultant Gröbner basis hold information of the target dynamical system. This framework using Gröbner basis can also represent availability of dynamical reconstruction by using delay embedding.

SUMMARY: The motion and spatial organization of subnuclear domains has been extensively studied by microscopy, but an understanding of the underlying mechanisms remains elusive. In this study, we used mathematical modeling and an in vitro cell system to investigate the mechanism underlying genomic architecture reorganization, a process commonly observed during cellular differentiation. We found that dynamic nuclear deformation provides a driving force for this reorganization by using a new approach of mathematical model based on multi-phasefield method. This study makes a significant contribution toward a better understanding of the mechanisms underlying the reorganization of nuclear architecture, a fundamental question of long-standing interest in genome and developmental biology.

SUMMARY: The proteases celled Kallikrein-related peptidase (KLKs) play critical role in epidermal desquamation. KLKs are expressed by keratinocytes in the stratum granulosum and secreted to the intercellular space within lamellar granules. Proteolytic activity induced by KLKs are essential for the desquamation process to occur since desquamation involves the degradation of corneodesmosomes which are the main adhesive structure in the stratum corneum. KLKs act on the protein components that comprise corneodesmosomes and the progress of corneodesmosome degradation loosens adhesion among corneocytes so that they shed off the surface of the stratum corneum. On the basis of experimental studies, we propose a mathematical model for the desquamation process.

SUMMARY: A model adaptive to three factors of angiogenesis, chemotactic gradient, chemotactic velocity, and chemotactic gradient variance, is realized by local activation and global inhibition.

SUMMARY: Cell deformation driven by the ECM degradation on plasma membrane is formulated as a Stefan-like free boundary problem which is reduced to a degenerate parabolic equation by Liouville’s transport theory.