/ Hiroyuki Sagawa / Professor
/ Hisasi Morikawa / Professor
/ Ken-ichi Funahashi / Associate Professor
/ Katsutaro Shimizu / 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 Method 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:
Refereed Journal Papers
The paper deals with boundary-value problems for the Laplacian and squared Laplacian in periodically perforated domains with homogeneous Dirichlet conditions on the boundaries of the holes. The period of perforation and the `size' of the hole with respect to the period 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.
In this paper, the asymptotic behavior of the solution of a mixed boundary-value problem for the Laplace operator in a domain with equal and periodically located stuck regions (with homogeneous Dirichlet data) is studied in two cases: the stuck regions are dispersed over the domain, or they are placed on the boundary. The period of the structure and the size of a stuck region compared with the period are small parameters. In the limit, the stuck regions disappear, and the formal limit problem (the averaged problem) does not necessary have solutions. In particular, this means that zero is an eigenvalue of the Laplace operator with corresponding boundary conditions. Several terms of the asymptotic expansion of the solution with respect to the small parameters are obtained. Since the limit problem in insoluble, the asymptotics constructed contain terms that increase unboundedly.
In this paper, we analyse mathematically the relationship between the mean field theory network (MFT) model and the continuous-time Hopfield neural network by the use of the theory of dynamical systems. This MFT model, which is obtained by applying the mean field approximation to the Boltzmann machine, is a discrete-time recurrent neural network. We prove that the set of asymptotically stable fixed points of the asynchronous MFT model coincides with the set of asymptotically stable equilibria of the continuous-time Hopfield neural network. Therefore, it is shown that the asynchronous MFT model is equivalent to the Hopfield neural network on the nature of the fixed points (or equilibria).
We propose a Monte Carlo method for solving the quantum many-body interacting systems. Mean fields dominating the structure of low-lying states are selected by a Monte Carlo method, which generate optimum many-body basis states for diagonalizing the hamiltonian consisting of one- and two-body terms. Not only the ground state but also low-lying excited states are obtained with their wave functions. Results are examined by comparison to exact values.
The amplitudes of isospin mixing to the ground states of $^{14}$O and several odd-odd N=Z nuclei are studied in Hartree-Fock approximation taking into account charge symmetry and charge independence breaking forces. We find that isospin mixing increases the sum rule of super-allowed Fermi transitions by 0.1% in $^{14}$O and more than 1% in $^{54}$Co because of core effects, while valence particles tend to decrease the sum rule. It is shown that the sum rule increase due to the CSB and CIB interactions amounts to about 40% of that due to the Coulomb interaction alone.
A new decay scheme between Gamow-Teller resonances and isobaric analog state by magnetic dipole transitions is studied. The sum rule of M1 transitions between IAS and GT states is found to be significantly enhanced compared to the non-energy weighted sum rule of the parent state. Calculated enhancement factors can be as large as $\sim$ 2.5 for $^{48}$Sc and $^{90}$Nb, and 1.5 for $^{208}$Bi. Transition strengths between specific states are calculated in Tamm-Dancoff approximation. The interest of measuring M1 transitions between IAS and GT states to obtain information on the spin-isospin response in finite nuclei is stressed.
Simple relations between the parameters of Skyrme-type interactions and the mean energies of isovector monopole, isovector dipole and Gamow-Teller resonances are derived in order to examine the isovector spin-independent and spin-dependent parts of the interaction. It is checked that these relations reproduce well the energies of RPA calculations using $^{208}$Pb as a test nucleus. We propose to include them as new, additional constraints when determining effective interactions to be used in neutron-rich or proton-rich nuclei.
A detailed understanding of complete fusion cross sections in heavy-ion collisions require a consideration of the effects of the deformation of the projectile and target. Our aim is to show that deformation and orientation of the colliding nuclei have a very significant effect on the fusion barrier height and on the compactness of the touching configuration. We analyze a few projectile-target combinations in a part of the superheavy region where $\alpha $ half-lives are calculated to be observable longer than 1 $\mu $s.
We study the effect of the unique shell-structure as well as the very low particle-threshold on collective modes in drip-line nuclei, first performing the Hartree-Fock (HF) calculation with Skyrme interactions and, then, using the random-phase-approximation (RPA) solved in the coordinate space with the Green's function method. We examine also one-particle resonant states in the HF potential. The properties of both isocalar and isovector monopole giant resonance (GR) are found to change drastically in nuclei around the neutron drip line. The characteristic feature of the isovector dipole modes as well as the isoscalar quadrupole modes in drip-line nuclei is also studied.
We consider increasing processes of class L, that is , increasing self-similar processes with independent increments. We decide a necessary and sufficient condition for the existence of positive increasing functions with liminf and limsup of the processes divided by them being 1 almost surely. Moreover we give a criterion to classify functions with liminf being 0 and infinity or limsup being 0 and infinity.
We consider multidimensional processes of Ornstein- Uhlenbeck type. We give a recurrence criterion of the process when its matrix is non-diagonalizable. Furthermore we compare it with the case when matrix is diagonalizable in two dimension.
In this paper, the asymptotic behavior of the solution of a mixed boundary-value problem for the Laplace operator in a domain with equal and periodically located stuck regions (with homogeneous Dirichlet data) is studied in two cases: the stuck regions are dispersed over the domain, or they are placed on the boundary. The period of the structure and the size of a stuck region compared with the period are small parameters. In the limit, the stuck regions disappear, and the formal limit problem (the averaged problem) does not necessary have solutions. In particular, this means that zero is an eigenvalue of the Laplace operator with corresponding boundary conditions. Several terms of the asymptotic expansion of the solution with respect to the small parameters are obtained. Since the limit problem in insoluble, the asymptotics constructed contain terms that increase unboundedly.
A new decay scheme between Gamow-Teller resonances and isobaric analog state by magnetic dipole transitions is studied. The sum rule of M1 transitions between IAS and GT states is found to be significantly enhanced compared to the non-energy weighted sum rule of the parent state. Calculated enhancement factors can be as large as $\sim$ 2.5 for $^{48}$Sc and $^{90}$Nb, and 1.5 for $^{208}$Bi. Transition strengths between specific states are calculated in Tamm-Dancoff approximation. The interest of measuring M1 transitions between IAS and GT states to obtain information on the spin-isospin response in finite nuclei is stressed.
We study the effect of isospin impurity on the super-allowed Fermi $\beta $ decay using microscopic HF and RPA (or TDA) model taking into account CSB and CIB interactions. It is found that the isospin impurity of N=Z nuclei gives enhancement of the sum rule of Fermi transition probabilities. On the other hand, the super- allowed transitions between odd-odd $J$=0 nuclei and even-even $J$=0 nuclei are quenched because of the cancellation of the isospin impurity effects of mother and daughter nuclei. An implication of the calculated Fermi transition rate on the unitarity of Cabbibo-Kobayashi-Maskawa mixing matrix is also discussed.
Theory of Complete fusion cross sections in heavy-ion collisions require a consideration of the effects of the deformation of the projectile and target. Our aim is to show that deformation and orientation of the colliding nuclei have a very significant effect on the fusion barrier height and on the compactness of the touching configuration. We analyze a few projectile-target combinations in a part of the superheavy region where $\alpha $ half-lives are calculated to be observable.