第5回数学総合若手研究集会

アブストラクト：プログラム

タイトル：A probabilistic algorithm which generates standard tableaux of a generalized Young diagram
アブストラクト：: In 1979, C.Greene, A.Nijenhuis, and H.S.Wilf construct a probabilistic algorithm which generates standard tableaux of a Young diagram. In this talk, we generalize this algorithm for a “generalized” Young diagram in the sense of D.Peterson and R.A.Proctor. This gives a proof of a hook length formula for the number of standard tableaux of a generalized Young diagram.

タイトル：Quantization of Maxwell field in Lorentz gauge
アブストラクト：: The quantized Maxwell field interacting with a charged nonrelativistic particle is considered in the Lorentz gauge. When the dipole approximation is made and a strong infrared convergent condition is imposed on the form factor, we can construct the quantized Maxwell field in an indefinite metric Krein space. We show that there exists the asymptotic field of the vector field, from which the (asymptotic)physical subspace is derived.

タイトル：A formula on logarithm of product of N exponential matrices was proved by Chaconand Fomenko. This formula is valid for time-1 maps of formal vector fields without constant term and linear term instead of exponential matrices. A time-1 map of a formal vector field without constant term and linear term is a formal diffeomorphism tangent to the identity map, and the formula gives a relation between those formal diffeomorphisms. Through hard calculations, we obtain concrete sufficient conditions, which are stated by characteristic curves associated with relations, to exist relations of two formal diffeomorphisms. By classifying characteristic curves satisfying one of the conditions, we construct examples of non commuting two formal diffeomorphisms.

タイトル：Large deviation and the tangent cone at infinity of a covering graph with the transformation group of polynomial volume growth
アブストラクト：: We discuss a large deviation property of a periodic random walk on a covering graph with its transformation group of polynomial volume growth in view of geometry. We observe that the behavior of random walk at infinity is closely related to the Gromov-Hausdorff limit of an infinite graph and in the case where the graph has action of group of polynomial volume growth, the Carnot-Carath¥'{e}odory metric shows up in its limit space.

タイトル：Rotation number approach to spectral analysis of the generalized Kronig- Penney Hamiltonians
アブストラクト：: In this talk, we discuss the spectrum of the one-dimensional Schr¥"odinger operator with periodic singular potentials. According to the Floquet-Bloch Theory, its spectrum consists of the infinitely many bands. Each consequtive bands are separated by an open set which is called the spectral gap. We give a characterization of the band edges of its spectrum by the rotation number.

タイトル：On the maximum value of ground states for the scalar field equation with double power nonlinearity
アブストラクト：: We consider uniqueness of positive solutions to semilinear elliptic equations. For semilinear equations, uniqueness of the solution is an important information to investigate more deep natures of the solutions. In this talk we review the history of this area and give some results by myself.

タイトル：Viewing quantum information theory from generic probability model (一般確率モデルから眺める量子情報理論)
アブストラクト：: Quantum information theory is an important and younger branch in information theory based on quantum mechanics. There have been proposed various mathematical approaches to such theory, not only the conventional one using Hilbert spaces. In this talk, I will introduce one such approach, called “generic probability model”, and from the viewpoint survey some topics in quantum information theory.

タイトル：Ricci flow on 3-dim Lie group and 4-dim Ricci flat manifolds
アブストラクト：: We study Ricci Flow on 3-dimensional Lie group and 4-dimensional Ricci flat manifolds. We construct 4-dimensional cohomogeneity one metrics behaving like a ricci flow.

タイトル：Properties of Monotone Infinitely Divisible Distributions
アブストラクト：: In monotone probability, we can define the notion of “infinite divisibility” as in the usual probability theory. I have found that the support of an infinite divisible distribution enjoys some interesting properties. For instance, the positivity of an infinitely divisible distribution is determined at one time (This is called a subordinator theorem in the usual probability). I will explain several support properties comparing with the usual probability.

タイトル：Phase portrait analysis for the envelope solution of cubic-quartic nonline Schro¨dinger equations (3 次-4 次非線形シュレーディンガー方程式に対する包絡解の相図解析と変調不安定性)
アブストラクト：: We study the behavior of the standing wave solutions for the one-dimensional cubic-quartic nonlinear Schr¥"odinger equation (CQNLS) $iA_t+pA_{xx}+q|A|^2A+¥varepsilon|A|^3 A=0$. The phase portrait analysis shows that the envelope solutions have a region in quartic nonlinearity, where both bright and dark solitons can coexist depending on the signs of $p$ and $q$.

タイトル：On the quasi-derivation relation for multiple zeta values (多重ゼータ値の一般導分関係式について)
アブストラクト：: Masanobu Kaneko conjectured that a generalization of certain derivation operators would give a new class of relations among multiple zeta values. The aim of this talk is to give a proof of the conjecture by reducing it to a class of relations for multiple zeta values studied by Gaku Kawashima.

タイトル：Heterodimensional tangencies on cycles leading to strange attractors
アブストラクト：: We study a two-parameter family of 3-dimensional dieomorphisms which have a bifurcation induced by simultaneous generation of a heterodimensional cycle and a heterodimensional tangency associated to two saddle points. We show that such acodimension-2 bifurcation introduces a quadratic homoclinic tangency associated to one of the saddle continuations and we detect nonhyperbolic phenomena arbitrarily close to the codimension-2 bifurcation.

タイトル：On the existence of the ground state and the asymptotic fields for the system of the Dirac field interacting with electromagnetic fields (ディラック場と電磁場が相互作用する系の基底状態の存在および漸近場の存在について)
アブストラクト：: In this talk we consider the Hamiltonian on the system of the Dirac field interacting with the electromagnetic field quantized in the Coulomb gauge. By introducing ultraviolet cutoffs and spatial cutoffs, it is seen that the Hamiltonian is a self-djoint operator on a boson-fock space. It is proven that the Hamiltonian has the unique ground state under the infrared regularity condition, and the asymptotic fields exist.

タイトル：The best constant of Sobolev inequality corresponding to 2-point boundary value problem for bending of a beam (弾性基盤上の張力をかけた棒のたわみの2 点境界値問題と対応するソボレフ不等式の最良定数)
アブストラクト：: We consider 2-point boundary value problem for bending of a beam. Its Green function is found. As an application, we found the best constant and thebest function of Sobolev inequality by investigating an aspect of Green function as are producing kernel.

タイトル：A generalized logarithmic module of hyperplane arrangements with multiplicities
アブストラクト：: A hyperplane arrangement is a finite collection of linear hyperplanes, and a multiplicity is a function from a hyperplane arrangement to non-negative integers. To these we can associate logarithmic derivation modules which play key roles in the theory of hyperplane arrangements. Recent developments of this area require us the existence of a generalized multiplicity and logarithmic module, i.e., all-integer valued one. We give a generalization of them and an application to free Coxeter multiarrangements.

タイトル：The verification of bifurcation of dynamics via the homology Conley index and rigorous computations
アブストラクト：: Recently the topological methods with rigorous computations are used to understand nonlinear phnomana. The homology Conley index, which is the topological invariant for invariant sets with some property, called isolated invariant sets, is one of the tool to detect fixed points, periodic orbits and chaos for systems. Here I present the new method and computer-assisted application to capture some bifurcations for ODEs or PDEs, so-called saddle-node, pitchfork and Hopf bifurcation by using the Conley index.

タイトル：Continuity points of typical functions
アブストラクト：: Since Banach and Mazurkiewicz independently proved that typical (in the sense of Baire category) continuous functions are nowhere differentiable, the study of the behaviour of typical continuous functions has been one of the most popular topics in classical real analysis. Despite being less popular, it is also interesting and important to investigate typical members of other families of functions. The talk will look at several families in which typical functions have small continuity points.

タイトル：Slip condition problem for inhomogeneous incompressible fluid-like bodies
アブストラクト：: Recently, granular materials have been studied by the use of continuum mechanics. Under certain conditions the response of them can be described as motion of inhomogeneous incompressible fluid-like bodies. Furthermore, it is well known that slip phenomena on the surface of the body play the significant role of its motion. Here, we will talk our mathematical results concerning an initial-boundary value problem of this model.

タイトル：Toy models for D.H.Lehmer’s conjecture (joint work with Eiichi Bannai)
アブストラクト：: In 1947, Lehmer conjectured that the Fourier coefficients of Ramanujan's $¥tau$-function are not vanish. In this talk, we consider that the elementary model of the Lehmer's conjecture. We show that the Fourier coefficients of the weighted theta series of $¥mathbb{Z}2$-lattice and $A_{2}$-lattice are not vanish. In other word, the spherical $5$ (resp. $7$)-design are not exist on the shells in the $?mathbb{Z}2$-lattice (resp. $A_{2}$-lattice).

タイトル：Finite-type invariants for curves, categorification of Jones polynomial with Z/2Z coefficients for knot, and word theory
アブストラクト：: This talk consists of two parts. In the first part of the talk, we reconstruct Arnold's invariants by using Turaev's theory of words and construct other invariants of curves. In the second part of the talk, on the joint work with T. Fukunaga (Hokkaido University) , we study about Jones polynomial where this polynomial is the knot invariant. We construct homology groups with grades (i, j) where Jones polynomial with Z/2Z coefficients is obtained by calculating Euler characteristic of the i-graded homology groups for each grade j fixed. This homology is strictly stronger than Jones polynomial with Z/2Z coefficients and independent of Jones polynomial with Z coefficients. We use Turaev's theory of words throughout the work.

タイトル：Division polynomials of hyper elliptic Jacobians
アブストラクト：: In the theory of elliptic curves, division polynomials are important for the study of the structure of the torsion subgroup. In this talk, we will generalize division polynomials to the case of hyper elliptic Jacobians over the complex numbers. We will also give determinantal expressions and some properties of division polynomials.

タイトル：Confrontations and concurrences of the original Navier-Stokes equations
アブストラクト：: Abstract. The two-constant theory is the one now accepted for isotropic, linear elasticity. The original Navier-Stokes equations or Navier equations were introduced in the prime of the second period of molecules in 1821. From the mathematical view of Navier-Stokes equations, we would like to report the theoretical confrontations and concurrences discussed then among Poisson, Navier, Cauchy, Saint-Venant, Stokes. This is not only the mathematical history, but also the modeling problem of functional equations, especially on the two-constant theory of isotropic elasticity and Navier-Stokes equations.

タイトル：Orbit shifted shadowing property of generalized Lozi maps
アブストラクト：: Coven, Kan and Yorke introduced the notation of the parameter-shifted shadowing property (for short PSSP). Kiriki and Soma proved that certain Lozi maps and geometric Lorenz maps have PSSP. We study a certain of shadowing property in which we shift some entries pseudo-orbits by rule instead of shifting parameter of maps, called the orbit-shifted shadowing property (for short OSSP). We will show that certain piecewise C^{2} Lozi-like maps (called generalized Lozi maps)introduced by Young have the OSSP.

タイトル：Some properties of time operators
アブストラクト：: We consider two operators $T$ and $H$ which satisfy the canonical commutation relation with one degree of freedom. In the quantum mechanical context, $T$ and $H$ are called time operator and Hamiltonian, respectively. In this talk, we study some examples of time operatoers and their properties.

タイトル：On the symmetry of the ground states of nonlinear Sch¥"odinger equation with potential
アブストラクト：: We consider the relation between the symmetry of the ground states of nonlinear Sch¥"odinger equation and the symmetry of the external potential. We show that sometimes the ground states does not have the symmetry which the external potential has.

タイトル：Homotopy classification of nanophrases and its application
アブストラクト：: In this talk I talk about the theory of words and its application to geometry. V.Turaevintroduced an equivalence relation of homotopy on the generalized phrases (called nanophrases). It is closely related to geometric objects. I will talk about some homotopy invariants of nanophrases and I give the homotopy classification of nanophrases with less than or equal to four letters without the condition on length. I will also talk about its application to topology of curves on surfaces.

タイトル：The strategic pricing of options : Black-Sholes equation (オプションの戦略的な価格付け: Black-Sholes 方程式の周辺)
アブストラクト：: This talk is about mathematical finance, option pricing. This study gives a example as a extention (extended Black-Sholes equation) : pays attention to a correlation among the stocks. On the other hand, this study constructs a game-theoretic model that the stock price depends on the player's strategic action. This study considers the relations between extended Black-Sholes equation and this model.

タイトル：The convergence with nondimensional parameter of Hasegawa-Wakatani equations
アブストラクト：: It is known that when we pass to the limit of nondimensional parameter of Hasegawa-Wakatani equations, we get Hasegawa-Mima like equations. In this presentation, we consider passing to the limit of nondimensional parameter of Hasegawa-Wakatani equations and we finally prove that there exists a strong solution for the initial boundary value problem for Hasegawa-Mima like equations.

タイトル：Existence of spike patterns for some chemotaxis models with saturation
アブストラクト：: We are concerned with stationary spike patterns for some chemotaxis models with saturation. For the purpose, we need solve some nonlocal problems. In this talk, we present some useful techniques to solve nonlocal problems.

タイトル：The existence of flat projective structures on Lie groups
アブストラクト：: I would like to determine the existence of left invariant flat projective structures on Lie groups. It is known that a left invariant flat projective structure on a Lie group corresponds to a certain Lie algebra representation. By using this correspondence I study the problem whether Lie groups admit flat projective structures or not. In this talk, I will review my preceding result about Lie algebras of dimension 5 or less, and introduce the progress.

タイトル：On a representation of a local gauge group
アブストラクト：: The local structure of the net of von Neumann algebra generated by a certain representation of a gauge group $C_c^{¥infty}(M,G)$ in a Boson Fock space is presented. Our approach is more algebraic than the preceding studies. Although the representation is infinite-dimensional, some of the Lie-group techniques are still available, since the gauge group is a nuclear Lie group. There are many differences between our gauge algebra and the algebra of free scalar fields, in particular there are no spectrum condition nor Reeh-Schlieder property in this algebra.

タイトル：Diffused interface model for premixed flame equation
アブストラクト：: We introduce a premixed flame equation, and study the limiting behavior of solutions to the equation. In particular, we approximate the equation appropriately for the dynamic interface. It is already known that the interfaces of solutions to Allen-Cahn equations approximates the motion of interfaces by the mean curvature flow equation. We show that the motion of diffused interfaces of premixed flame equation have the similar behavior to those of the Allen-Cahn equation.

タイトル：Rotational symmetry applied to boundary element analysis (回転対称性を考慮した境界要素法)
アブストラクト：: This work is to develop an efficient boundary element technique for solving a potential problem with a geometry having rotational symmetry. When assuming an n-fold rotational symmetry, the system matrix becomes a cyclic one, with the result that the matrix size is reduced to 1/n^2. Results of test calculations demonstrate the validity of the present formulation. The author’s research group has a plan to apply this simplified method to analyze fusion plasma that has a helical geometry.

タイトル：On the structure of certain K3 surfaces
アブストラクト：: I will explain the structure theorem for elliptic K3 surfaces with two singular fibers both of type $¥rm{I}_4^*$. These K3 surfaces are classified into two classes. Every surface which belong to the one admits a Shioda-Inose structure. And general member of the other class does not admit a Shioda-Inose structre, but is isomorphic to a Kummer surface of product type.

タイトル：Plasma diffusion due to uncertainty
アブストラクト：: Quantum mechanical plasma diffusion is studied using a semi-classical model with two different characteristic lengths; one is the average interparticle separation, and the other is the magnetic length. The diffusion coefficients D derived in this study are qualitatively similar dependence to experiments on several plasma parameters, such as the temperature T, the mass m, the density n, and the magnetic field B. The numerical values of the diffusion coefficient D, in this study are as large as that of the neo-classical diffusion.

タイトル：Artificial neural network for reactor stability monitoring (人工ニューラルネットワークによる原子炉安定性診断)
アブストラクト：: In boiling water reactor stability monitoring, damping ratio has been used as a stability index. A method for estimating the damping ratio by applying Principal Component Analysis to neutron detector signals measured with local power range monitors had been developed. The self-organizing map(SOM) is one of the artificial neural networks(ANNs) and has the characteristics such that online learning is possible without supervised learning. In the present study, the SOM was applied to extract the relevant signal components more quickly and more accurately for online monitoring, and the availability was confirmed through the feasibility study. And for realizing online stability monitoring only with ANNs, another type of ANN that performs online processing of PCA was combined with SOM. And stability monitoring performance was investigated.

タイトル：Spacelike hypersurfaces and spacelike submanifolds in de Sitter space
アブストラクト：: We introduce the notion of lightcone Gauss maps and lightlike hypersurfaces to investigate the differential geometry of spacelike hypersurfaces and spacelike submanifolds in de Sitter n-space. Singularities of these maps have relationships with the geometrical properties of spacelike submanifolds. If n is small, the singular types of generic Gauss maps and generic lightlike surfaces can be classified.

タイトル：Random walks on finite groups
アブストラクト：: We consider a simple random walk over the vertices of an m-gon. It is well known that such a random walk converges to the uniform distribution if m is odd. What is the optimal number of moves needed to have this convergence? An answer of this question and some similar results for even integers m are presented.

タイトル：Gauss maps of hypersurfaces in Euclidean n-sphere and duality
アブストラクト：: We consider Gauss maps of hypersurfaces in Euclidean n-shpere by using Legendrian singularity theory. The pair of the hypersurface and Gauss map is a Legendrian immersion from source of hypersurface to a contact manifold. We can interpret the image of Gauss maps as the wavefront set of this Legendrian immersion. We show that the generating family of this Legendrian immersion is the height function on the hypersurface, we classify singularities of Gauss map by using this viewpoint. We also consider the relations with principal curvature of the hypersurfaces and that of Gauss maps by using duality of spheres.

タイトル：Algebraic method for multibody problem
アブストラクト：: The algebraic model (ALG) proposed by the authors has sufficiently high accuracy in calculating the motion of a test particle with all the field particles at rest. When all the field particles are moving, however, the ALG has poor prediction ability on the motion of the test particle initially at rest. None the less, the ALG approximation gives a good results for the statistical quantities, such as variance of velocity changes or the scattering cross section, for a sufficiently large number of Monte Carlo trials.

タイトル：On tight closure with a view point from algebraic geometry
アブストラクト：: Tight closure is an ideal operator on a ring of positive characteristic. Its definition is purely algebraic. However, objects and singularities in tight closure have a connection with those in algebraic geometry. I introduce the connection and basic related topics. If there is a space, I also introduce my result.

タイトル：Characterization of certain plane curves by Galois point
アブストラクト：: In 1996, H. Yoshihara introduced a new notion “Galois point” for a plane curve, in algebraic geometry. As an application of studying Galois point, we can characterize certain plane curves by using Galois point. I will present such characterization results.

タイトル：Three-dimensional Cauchy condition surface method to reconstruct the fusion plasma boundary shape (3 次元コーシー条件面法による核融合プラズマ境界形状の逆推定)
アブストラクト：: The 2-D Cauchy condition surface (CCS) method to reconstruct the plasma boundary shape has been expanded to deal with a 3-D fusion plasma. This 3-D CCS method solves the 3-D Poisson-type equations in terms of vector potential with the aid of measured magnetic sensor signals. Results obtained in a trial calculation for axisymmetric plasma indicate that the new 3-D method can also determine the plasma boundary shape accurately in the same way as the 2-D CCS method.

タイトル：An introduction to exterior differential system
アブストラクト：: We study exterior differential system (EDS). An EDS is a system of equations on a manifold defined by equal to zero some differential forms. It is one of our concerns when integral manifolds for given EDS exists, i.e. submanifolds satisfying the system of equations. In this poster session, I introduce Cartan-K¥"ahler theorem and its application. This theorem is fundamental existence result for integral manifold of theorem for analytic PDE.

タイトル：Time averaged properties along unstable periodic orbits in some systems of ordinary differential equations
アブストラクト：: By employing low dimensional chaotic systems described by ordinary differential equations, we collected many unstable periodic orbits(UPOs) for each system numerically. Then we study time averaged properties of a set of UPOs classified by the period compared to those of a set of segments of chaotic orbits with the corresponding lengths.

タイトル：Stochastic process in quantum mechanics via two-state vector formalism
アブストラクト：: In quantum mechanics, the measurement outcome is given by the Born formula but its probabilistic measure cannot be well-defined in the conventional probabilistic theory discussed by Nelson et al. Extending the concept of the measurement outcome, we consider an experimental accesible quantity as a weak value initiated by Aharonov et al. related to the two-state vector formalism. Following the Baysian rule, we can consider weak values as the complex conditional probability to well-define the probabilitic measure. We can discuss the stochastic process in quantum mechanics based on the extended measurement theory. This work is with Prof. Akio Hosoya.

タイトル：Inverse scattering problems for nonlinear dispersive equations
アブストラクト：: We discuss inverse scattering problems for nonlinear dispersive equations. The nonlinearity of the equations is a non-local term with unknown interaction potentials. We determine the unknown interaction potentials and reconstruct the nonlinearity by the knowledge of the scattering states.

第5回数学総合若手研究集会
〜他分野との学際的交流を目指して〜

- The 5th Mathematics Conference for Young Researchers -

Contents

アブストラクト：プログラム

第5回数学総合若手研究集会〜他分野との学際的交流を目指して〜

- The 5th Mathematics Conference for Young Researchers -

Contents

アブストラクト：プログラム

第5回数学総合若手研究集会
〜他分野との学際的交流を目指して〜