/ Hiroyuki Sagawa / Professor
/ Hisasi Morikawa / Professor
/ Nguyen Van Giai / Visiting Professor
/ Peter Möller / Visiting Professor
/ Ken-ichi Funahashi / Associate Professor
/ Katsutaro Shimizu / Associate Professor
/ Gleb V. Nosovskij / Visiting Associate Professor
/ A. G. Belyaev / Visiting Researcher
/ Sergei Duzhin / Visiting Researcher
/ Kazuto Asai / Assistant Professor
/ Michio Honma / Assistant Professor
/ Shigeru Watanabe / Assistant Professor
/ Toshiro Watanabe / Assistant Professor
/ Hiroshi kihara / Research Associate
The scope of activities of the Center for Mathematical Sciences spans all aspects of education and research in the fields of mathematical sciences. Current research directions in the field of mathematics are joined by the common theme "Geometrical Methods in Mathematical Sciences". In the fields of physics, theoretical research is performed in many-body theories, nuclear physics and quantum gravity. Together with this, there is a project to develop educational software on quantum physics. The research areas assigned to each co-researcher are as follows:
Prof. H. Morikawa investigates the theory of geometrical transformations, the algebraic methodology.
Prof. H. Sagawa studies the physics of the system of several subjects including nuclei and microclusters.
Prof. Nguyen Van Giai studies quantal many-body problems of metal clusters and nuclear physics.
Prof. P. Möller is working on super heavy elements and also developing 3D graphics and software for physics education.
Prof. K. Funahashi researches the theory of the neural networks and function theory of several complex variables from a geometrical viewpoint.
Prof. K. Shimizu advances the traditional quantum theory and creates the geometrical theory of quantum gravity.
Prof. T. Watanabe generalizes the unimodality problems of 1-dimensional infinitely decomposable distributions to multi-dimensional cases in the use of geometrical methods as the theory of manifolds.
Prof. K. Asai researches combinatorial identities for geometrically generalized tableaux, and the connection between graphical compositions of indeterminate functions of several variables and homogeneous P.D.Es. with constant coefficients.
Prof. S. Watanabe studies geometrical interpretations of generating functions for spherical functions on homogeneous spaces.
Prof. M. Honma researches the microscopic structures and dynamics of the nuclei by algebraic methods and geometrical models performing the quantitative analysis by large-scale numeric calculations.
Prof. H. Kihara studies higher dimensional differential topology and elucidates the role of higher dimensional phenomena in various areas of mathematical sciences.
Prof. S. Duzhin unifies algebraic topology, differential geometry and combinatorics with his investigations of nonlinear differential equations, singularity theory and knot invariants.
Prof. G. Nosovskij is interested in nonlinear partial differential equations and theories of probability. In particular, he studies the theory of Hamilton-Jacobi-Bellman equations which is applied to the modern theory of the stochastic process.
Prof. A. Belyaev studies homogenization theory for partial differential equations in media with periodic structures. He is also interested in applications of methods of differential geometry and partial differential equations in shape and image analysis.
Refereed Journal Papers
The paper deal with boundary-value problems for Laplace and bilaplace operators in periodically perforated domains with homogeneous Dirichlet conditions on the boundaries of the holes. The period of the perforation and the `size' of the hole with respect to the period of the perforation are two small parameters. Asymptotic behavior of the solutions, eigenvalues, and eigenfunctions of the boundary-value and spectral problems are investigated for various relations between the small parameters.
We propose a new approach for recognition, description and extraction of ridges and ravines based on singularity theory; the approach may be used for shape coding and animation.
A new concept of ridges, ravines and related structures (skeletons) associated with a surface in three-dimensional space is introduced. The concept is based on the investigation of relationships between the singularities of the distance function from the surface and singularities of caustic of the normals to the surface. It involves both local and global geometric surface properties and leads to the effective procedure for ridges and ravines recognition.
Boundary value problems with oscillating boundary conditions are considered. Asymptotic behavior of the solutions is investigated in the case when the corresponding limit problems lie on their spectra.
To analyze given object shapes, it is necessary first to model the shapes and then to analyze the models. This paper proposes a method of modeling and analyzing two-dimensional (2D) and three-dimensional (3D) shapes based on singularities. First, a function is defined on an object. The object is then modeled by the distribution of hte singularities of the function. Finally, the extracted singular points are analyzed by a Reeb graph together with multiresolution analysis. The applications of the method include analysis of botanical leaf shapes and human facial expressions.
The composition of the random-phase-approximation ground state in metal clusters is explicitly determined. It is shown that, due to the long-range character of the Coulomb interaction, multiconfigurational, higher-order correlations in the ground state play a more important role than had been previously assumed. However, the deviations between single-particle occupation probabilities associated with the correlated ground state and the uncorrelated reference determinant are found to be small, thus validating the quasi-boson approximation. The reliability of various many-body approaches in describing excited collective modes depends crucially on their ability of adequately approximating these correlations in the composition of the ground-state wave function. In this context, the consistency of random-phase-approximation and configuration-interaction results in metal clusters is assessed through a comparison with established experimental trends and known theoretical limits of the exact many-body problem.
The energy-density functional approach and jellium-like models are used to examine two important electronic properties of metal (Li, Na, K) clusters: their shell and supershell structures, and the behaviour of plasmon energies with increasing cluster sizes. A comparative study is made between predictions of the usual jellium model and those of the pseudo-jellium model where pseudohamiltonians are used.
Decay of soft multipole excitations in halo nuclei is studied in comparison with the potential resonance state. Although the soft excitation has a sharp peak just above the particle threshold and carries extremely large transition strength, the decay rate looks much faster than that expected for a resonance state. Consequently, the half life is shown to be several orders of magnitude shorter than what one naively expects from the uncertainty principle. It is shown also that the soft excitations accumulate large transition strength as a non-resonant single-particle excitation, but not as particle-hole collective excitations like giant resonances.
A systematic study of the dipole plasmons and polarizabilities of potassium clusters within the pseudojellium model and using the TDLDA is presented. We explicitly show that effects of core electrons can explain the discrepancy between the jellium model and experiments.
Using realistic pseudohamiltonians and pseudopotentials to describe ion-valence electron interaction, we investigate the effects of non-locality of ion-electron interaction on the bulk plasmon energy in alkali metals. It is found that pseudohamiltonians and pseudopotentials can lead to important effects on the optical mass and plasmon energies.
In this paper we study the geometric topology of q-spherical fiber spaces over the n-sphere. The main theorem states a necessary and sufficient condition that such a space has the homotopy type of a topological(or smooth) manifold. This theorem gives simple examples of the differences of the homotopy types of Poincare complexes, topological manifolds and smooth manifolds.
Excitation energies, branching ratios, and E2/M1 mixing ratios of the negative-parity states in $^{125,127}$Xe and $^{125,127,129}$Cs have been calculated by using the Interacting Boson Fermion Model. A fairly good agreement with the experiment has been obtained. The sensitivity of the calculated observables to changes in model parameters is discussed. The results are compared with those obtained from the Triaxial Rigid Rotor plus Particle Model.
The transverse and the longitudinal momentum distributions of projectile fragmentation of unstable nuclei are studied by using a peripheral direct reaction model. We found that the transverse momentum distribution is affected by absorptive cutoff of the fragmentation process, and the width becomes narrower than that of the longitudinal one which remains almost unaffected. The momentum distributions of fragments from unstable projectiles $^{11}$Be and $^{11}$Li are studied for various microscopic wave functions. Calculated results with halo wave functions show good agreement with experimental data. We discuss also the difference among various models, the spectator model, Friedman model and Serber model, for predicting the momentum distribution.
The low-lying positive parity states of $^{11}$Be are calculated in a coupled channels treatment of a valence neutron interacting with a deformed core. The loosely bound nature of the valence neutron is taken into account by using a Woods-Saxon potential. Comparisons are made to shell-model predictions and to data. The model reproduces the measured spectrum quite well for realistic parameters of the neutron-core interaction.
Decay of soft multipole excitations in halo nuclei is studied in comparison with the potential resonance state. Although the soft excitation has a sharp peak just above the particle threshold and carries extremely large transition strength, the decay rate looks much faster than that expected for a resonance state. Consequently, the half life is shown to be several orders of magnitude shorter than what one naively expects from the uncertainty principle. It is shown also that the soft excitations accumulate large transition strength as a non-resonant single-particle excitation, but not as particle-hole collective excitations like giant resonances.
We investigate the use of elastic and inelastic scatterings with secondary beams of radioactive nuclei as a mean to obtain information on ground state densities and transition matrix elements to continuum states. An eikonal model is developed for this purpose by using the folding potential. In particular we discuss possible signatures of halo wavefunctions in elastic and inelastic scattering experiments.
Let $X$ be a non-compact Riemannian symmetric space of rank 1. Then it is known that $X=G/K$, where $G$ is a connected simple Lie group with finite center and $K$ is a maximal compact subgroup of $G$, and if $G=KAN$ is an Iwasawa decomposition, we have $\dim A=1$. From the classification theory, it is known that $X$ is either one of the classical hyperbolic spaces or the exceptional space. Let $M$ be the centralizer of $A$ in $K$, then the Martin boundary $K/M$ of $X=G/K$ is not a symmetric space, except for the case of real hyperbolic spaces. But as is well known, $(K,M)$ is a Gelfand pair. In the case of real hyperbolic spaces, we have the classical theory of spherical harmonics ; the zonal spherical functions are given essentially by the Gegenbauer polynomials and we have the classical generating function expansion for them. On the other hand, we showed that similar constructions are possible also in the other classical cases. The purpose of this paper is to give a construction similar to those of the classical cases for the exceptional case.
Let $G$ be a classical connected simple Lie group of real rank 1. And let $G=KAN$ be an Iwasawa decomposition of $G$ and $M$ be the centralizer of $A$ in $K$. Then $G/K$ is the the classical hyperbolic space. We denote by $D$ the Laplace -Beltrami operator on $G/K$ and by $P$ the Poisson kernel. For any complex number $s$, the function $P^{s}$ is an eigenfunction of $D$. Now we consider the converse. This problem is solved in the affirmative for the real hyperbolic space. The purpose of this paper is to give a quick proof in the real case using the spherical functions on $K/M$. Also included are some preliminary results about this problem, which might be useful to solve the problem in other cases.
V. Bargmann showed that a generating function for the system of the Hermite polynomials can be regarded as the integral kernel of a unitary mapping from an $L^{2}$ space onto a Hilbert spaces of analytic functions. Then we have the following problem : is a similar construction possible for any system of classical orthogonal polynomials ? That is, for any system of orthogonal polynomials, can we construct its generating function which can be regarded as the integral kernel of a unitary mapping from an $L^{2}$ space onto a Hilbert space of analytic functions ? The purpose of this paper is to show that this problem can be affirmatively solved for the system of the spherical functions on the homogeneous space $U(n)/U(n-1)$.
A d-dimensional OU type process X is a Markov process obtained from a spatially homogeneous Markov process undergoing a linear drift force determined by a matrix -Q. A recurrence criterion for X is given by means of the matrix Q and the Levy measure of X in case Q has diagonal Jordan cannonical form with distinct positive eigenvalues. This result extends results of T. Shiga.
We introduce a new concept of ridges, ravines and related structures (skeletons) associated with surfaces in three-dimensional space that generalizes the medial axis transformation approach. The concept is based on singularity theory and involves both local and global geometric properties of the surface; it is invariant with respect to translations and rotations of the surface. It leads to a method of hierarchic description of surfaces that yields new approaches to shape coding, rendering and design. The extraction of the features is based on differential geometry of surfaces with consequent segregation via multiscale analysis. Terrain feature recognition, dental shape reconstruction and medical imagery are a partial list of applications.
Many medical, geographical, and computer vision applications require the automatic registration of the 3D images of the same original surface as seen from different directions. For that reason we need to implement visual invariants reflecting essential surface properties. In this paper we investigate view- and coordinate-independent ridges, ravines, and related surface point features; they turn out to be closely connected with singularities of wavefronts, caustics, and Euclidean distance skeletons and, therefore, involve both local and global surface properties. We present methods for their extraction based on advanced differential geometry and demonstrate applicability of the introduced features in image understanding. The analysis of CT images, terrain feature recognition, and fashion design are a partial list of applications.
Analyzing the 3D images of a given surface as viewed from different positions naturally leads to investigation of coordinate-independent geometric surface features reflecting its essential properties. In the present paper we study surface point features related to ridge and ravine lines on a surface. These lines introduced in our previous works are defined as curves corresponding to the boundary points of the skeleton of the distance transform of the surface. We show that the ridges and ravines can be extracted via the directional derivatives of the principal curvatures along the associated principal directions. However, even after this local description the direct extraction of the ridges and ravines is a time-consuming procedure. It turns out that the ridge and ravine lines contain some remarkable points (end points and others) that can be extracted relatively easily. After finding such points the procedure of ridge and ravine extraction becomes much simpler. Moreover, these points are closely connected with some singularities of caustics and wavefronts, and have an independent interest in image analysis as visual invariants. The paper is devoted to the investigation of such points and the accompanying geometry of singularities of wavefronts and caustics.
We reviewed several approximation theorems in neural network theory. Especially, we applied an approximation theorem to the analysis of time series prediction using feedforward neural networks and proved that the neural network method is better than the linear prediction generically in the case of non-Gaussian processes.
We proved that any finite time trajectory of a given n-dimensional dynamical system can be approximately realized by the internal state of the output units of a continuous time recurrent neural network with a given time constant, n output units, some hidden units and an appropriate initial condition. As a corollary, we also showed that any continuous curve can be approximated by the output of a recurrent neural network.
The structure of the superdeformed states is studied from the viewpoint of collective nucleon pairs. The interacting boson model is extended to describe the superdeformation. A phenomenological hamiltonian is presented and applied to clarify the stabilization mechanisms and the spin dependent properties of the superdeformed states. It is shown that the unified description of two different configurations, normal low-lying states and the superdeformed states can be made within this framework.
We discussed several structure problems of nuclei both near the neutron and the proton drip lines. The conditions for the growth of halo nucleus are discussed together with experimental evidence of neutron and proton halos. Next, a microscopic Hartree-Fock (HF) + random phase approximation (RPA) model is applied to calculate dipole excitations in halo nuclei, and the results are compared with experimental data in $^{11}$Be. It is pointed out that the abnormally large dipole strength is accumulated by the threshold effect inherent to halo nuclei. Giant Gamow-Teller (GT) states in nuclei near the proton drip line is also studied by using the same microscopic model. The nuclei with Z$>$28 are predicted to have giant GT beta decays which carry most of the sum rule strength.
We study the isospin impurity problem in nuclei near the proton drip line. The probabilities are obtained from the calculated sum of the Fermi transition probabilities of N=Z even-even nuclei, which is strictly forbidden if the isospin is a good quantum number. We assume that the T=1 component is only mixed into the ground state of N=Z nuclei and no T$>$1 components exist. The mixing probabilities are 0.2 % in $^{16}$O and 0.6 % in $^{40}$Ca. It goes up quickly when the proton number is increased: 3% in $^{80}$Zr and 4% in $^{100}$Sn.