/ Hisasi Morikawa / Professor
/ Hiroyuki Sagawa / 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
/ 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. The Center for Mathematical Sciences and the group of mathematicians is headed by Prof. H. Morikawa, the author of numerous classical results in algebraic geometry and invariant theory. Current research of the Center in the field of mathematics is done in several directions, joined by the common title ``Geometrical Method in Mathematical Sciences''. In the fields of Physics, theoretical research is performed in 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. H. Morikawa investigates the theory of geometrical transformations, the algebraic methodology. H. Sagawa studies on the physics of the system of several subjects including nuclei and microclusters. K. Funahashi researches the theory of the neural networks and function theory of several complex variables from a geometrical viewpoint. K. Shimizu advances the traditional quantum theory and creates the geometrical theory of quantum gravity. 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. 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.E. with constant coefficients. S. Watanabe studies geometrical interpretations of generating functions for spherical functions on homogeneous spaces. M. Honma researches the microscopic structures and dynamics of nuclei or finite systems of quantum many-bodies by the algebraic methods and geometrical models performing the quantitative analysis by the large-scale numeric calculation. H. Kihara studies higher dimensional differential topology and elucidates the role of higher dimensional phenomena in various areas of mathematical sciences. S. Duzhin unifies algebraic topology, differential geometry and combinatorics with his investigations of nonlinear differential equations, singularity theory and knot invariants. G. Nosovskij is interested in nonlinear partial differential equations and theory of probability. In particular, he studies the theory of Hamilton-Jacobi-Bellman equations which is applied to the modern theory of stochastic processes. A. Belyaev applies geometrically the theory of homogenization to partial differential equations in media with periodic and random structures.
Refereed Journal Papers
Elliptic boundary value problems are considered in periodically perforated domains. Asymptotic behaviors of solutions as the period of perforation and certain geometrical parameters tend to zero are investigated
We give a short proof of an elegant
formula due to E. A. Gaydar which expresses the greatest common divisor of a
set of multivariate polynomials through the elements of the Grbner basis
of the ideal generated by these polynomials.
We give a short introduction to the
theory of Vassiliev knot invariants - a notion that incorporates all known
knot invariants. Then, we prove that the number of independent Vassiliev knot
invariants of order is less than
- thus strengthening the a priori bound
In this paper, we prove 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 n output units, some hidden units, and an appropriate initial condition. The essential idea of the proof is to embed the n-dimensional dynamical system into a higher dimensional one which defines a recurrent neural network. As a corollary, we also show that any continuous curve can be approximated by the output of a recurrent neural network.
In this paper we study the geometric topology of q-spherical fiber spaces over the n-sphere. The fiber homotopy equivalence classes correspond to the elements of the (n-1)-stable homotopy group of spheres. So we describe the necessary and sufficient condition that a q-spherical fiber space over the n-sphere has the homotopy type of a topological(or smooth) manifold in terms of its representative in the stable homotopy group. Then we can give simple examples of the differences of the homotopy types of Poincare complexes, topological manifolds and smooth manifolds.
Attempts to explain the source of
-process elements in Nature by particular astrophysical sites face the
entwined uncertainties stemming from the extrapolation of nuclear properties
far from stability, inconsistent sources of different properties (e.g.,
nuclear masses and half-lives) and the (poor) understanding of astrophysical
conditions, which are hard to disentangle. We utilize the full isotopic
-process abundances in nature [especially in all of the three peaks
(
, 130, 195)] and a unified model for all nuclear properties
involved (aided by recent experimental knowledge in the
-process path), to
deduce uniquely the conditions necessary to produce such an abundance
pattern. Recent analysis of a few isotopic ratios in the
and
-process peaks led to the conclusion that the
-process
abundances originate from a high-density and high-temperature environment,
which supports an equilibrium between neutron captures and
photodisintegrations. This excludes events where neutrons are released from
reactions in explosive He burning. The present study
investigates also the nature of the steady flow equilibrium of beta decays
between isotopic chains. We find strong evidence that steady flow was not
global but only local in between neighboring peaks, which requires time
scales not much bigger than 1 s. The abundances have to be explained by a
superposition of
-process components with varying neutron number densities
cm
and temperatures
K, where each of the
components proceeds up to one of the peaks. The remaining odd-even effects in
observed abundances indicate that neutron densities dropped during freeze-out
by orders of magnitude on time scales close to 0.04 s. A set of
-
conditions is presented as a test for any astrophysical
-process
site. We also show how remaining deficiencies in the produced abundance
pattern can be used to extract nuclear properties far from stability.
We review several important
experimental and theoretical developments that during the past decade have
revived interest in the stability properties of the heaviest elements. On the
experimental side two accomplishments stand out. One is the extension of the
known elements to Ns,
Hs and
Mt. The other is the
collection of an extensive body of data on the transition between asymmetric
and symmetric fission in the region close to proton number
and
neutron number
. On the theoretical side it has become clear
that some models that appropriately account for the most important
nuclear-structure aspects are sufficiently reliable for meaningful
applications to new regions of nuclei and to studies of new phenomena.
We tabulate the atomic mass excesses
and nuclear ground-state deformations of 8979 nuclei ranging from O to
. The calculations are based on the finite-range droplet macroscopic
model and the folded-Yukawa single-particle microscopic model. Relative to
our 1981 mass table the current results are obtained with an improved
macroscopic model, an improved pairing model with a new form for the
effective-interaction pairing gap, and minimization of the ground-state
energy with respect to additional shape degrees of freedom. The values of
only 9 constants are determined directly from a least-squares adjustment to
the ground-state masses of 1654 nuclei ranging from
O to
106
and to 28 fission-barrier heights. The error of the mass model is 0.669 MeV
for the entire region of nuclei considered, but is only 0.448 MeV for the
region above
.
A formalism suitable for numerical evaluation of finite-range liquid-drop model expressions related to the interaction energy of two arbitrarily oriented, deformed heavy ions is developed. The presentation of the formalism is organized to facilitate extensions to alternative parameterizations and energy expressions. The model is applied to specific heavy-ion collisions that illustrate the importance of a multidimensional approach in the study of complete fusion reactions. Potential-energy surfaces related to light-particle emission for heavy, deformed nuclei are also presented.
Energy spectra and electric dipole
transitions of N=7 isotones are studied by shell model calculations with
isospin-dependent kinetic energies for sd-shell orbits. Our model is designed
to reproduce the parity-inverted spectra near the ground state of Be.
The ground states of
Li and
He are predicted to have abnormal
parities as a result of a smooth isospin dependence of kinetic enegies.
Electric quadrupole moments in light
mirror nuclei are studied by shell model calculations with the proton-neutron
formalism. Our calculations describe successfully the Q-moments of both
loosely-bound and well- bound nuclei. The adopted effective charges are
consistent with the theoretical predictions due to the core polarization
effect. The large enhancement in B and
F shows a clear evidence
of proton halos.
We performed relativistic mean field calculations systematically in various mass region and compared with non-relativistic H-F calculations
Differential cross sections and vector
and tensor analyzing power for the Be (d,
He )
Be reactions
at 70 MeV have been measures and Gamow-Teller spinflip dipole transitions are
identified. Observed mean excitation energy and strength for the spinflip
dipole transition in which possible neutron halo effect manifests itself are
consistent with the shell model prediction which takes the neutron halo into
account.
The differential cross sections of
elastic and inelastic channels are studied by the optical potential model
with microscopic form factors. Our model reproduces the inelastic cross
sections to giant resonances in Pb with C projectiles. The cross
sections of soft multipole excitations are predicted by using the same model.
Performing Hartree-Fock plus RPA (TDA)
calculations, it is shown that ``superallowed ''Gamow-Teller beta decays are
possible for light neutron drip line nuclei without calling for the phenomena
related to a neutron halo. The calculated results are in agreement with the
feature of the observed GT strength in He. The result calculated by
using a schematic model with simple separable interaction is also presented.
A new framework, the variational shell
model, is proposed to describe the structure of neutron-rich unstable nuclei.
An application to Be is presented. Contrary to the spherical
Hartree-Fock model, the anormalous 1/2
state and its neutron halo are
presented with the Skyrme (SIII) interaction.
Effects of CIB(charge independence
breaking ) and CSB(charge symmetry breaking) forces on the Coulomb
displacement energies of isobaric analog states are investigated for
Ca,
Zr and
Pb. The mass number dependence of the
Coulomb energy anomaly is well explained when we employ CIB and CSB
interactions which reproduces the differences of both the scattering lengths
and the effective ranges of n-n scatterings.
The parity inversion anomaly in the
spectra of N=7 isotones is discussed based on contemporary shell model wave
functions. It is shown that the proton-neutron monopole interaction in the
mean field has a large effect on the energy gap between the 1/2 and 1/2
states in
Be , but not enough to explain the inversion spectrum.
We point out that the effects of quadrupole core excitation and pairing
blocking are equally important to make the parity inversion in
Be.
We discuss the effect of breakup on
heavy-ion fusion reactions induced by a halo nucleus Li based on a
semiclassical method. Our formula leads to a smaller effect of breakup than
that calculated in a recent paper using unitary. We show that although the
large enhancement of the fusion cross section than the other Li isotopes.
This trend is especially significant at low enegies.
It is shown that ``superallowed''
Gamow-Teller (GT) beta decays are possible for N=Z even-even nuclei heavier
than Ni, performing Hartree-Fock plus random phase approximation (or
Tamm-Dancoff approximation) calculations. Since Fermi-type beta decays are
forbidden in N=Z nuclei, it is pointed out that the main decay mode of the
nuclei such as
Sn may be superallowed GT beta decay, which is a beta
decay to the GT giant resonance state.
The signature inversion in odd-odd nuclei is studied in the interacting boson-fermion model (IBFM) . Energy levels and electromagnetic matrix elements are calculated with the SU(3) and the O(6) IBM2 core. When the particles occupy high-j orbits , the signature inversion occurs with the O(6) core by strong exchange interactions between fermions and the O(6) core.
It is known that there is a Levy process whose path is oscillating as time increases. But as far, it is considered that the mode of a unimodal Levy process cannot behave complicatedly. Also, it is known that self-decomposable Levy processes are unimodal. In this paper, it is shown that there exists a unimodal Levy process with oscillating mode, that is, all nonsymmetric self-decomposable and semi-stable Levy processes with negative parameters have oscillating modes.
In this paper, the author answers some questions raised by K.Sato by giving concrete examples. Namely, it is shown that there is a nonsymmetric unimodal Levy process which is the independent difference of unimodal and non-unimodal increasing Levy processes, and that there is a unimodal Levy process which is the independent sum of non-unimodal Levy process and Brownian motion. Further, the author gives an interesting example of a Levy process which repeats unimodality and non-unimodality infinitely many times as time increases or decreases.
Unrefereed Papers
Technical Reports
Academic Activities
Invited Professor at the Universite de Saint Etienne.
A reviewer for Mathematical Reviews.