/ Hiroyuki Sagawa / Professor
/ Noriaki Kamiya / Professor
/ Hisasi Morikawa / Professor
/ Ken-ichi Funahashi / Associate Professor
/ Katsutaro Shimizu / Associate Professor
/ 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. Our current researches in the field of mathematics are devoted to various subjects and problems arising in both pure and applied mathematics. In the fields of Physics, theoretical research is performed in many-body theories, Nuclear Physics and Quantum Gravity. There has been also a project to develop educational textbooks and software for mechanics, electromagnetism and quantum mechanics. The research areas assigned to each co-researcher are as follows:
Refereed Journal Papers
A self-consistent particle-phonon coupling model is used to investigate the properties of the isobaric analog resonance in $^{208}$Bi. It is shown that quantitative agreement with experimental data for the energy and the width can be obtained if the effects of isospin-breaking nuclear forces are included, in addition to the Coulomb force effects. A connection between microscopic model predictions and doorway state approaches which make use of the isovector monopole resonance, is established via a phenomenological ansatz for the optical potential.
We have studied effects of tensor correlations and couplings to 3$\hbar \omega $ excitations on the quenching of spin-dipole strengths in $^{12}$C and $^{16}$O by large scale shell model calculations. We found that the transition strengths in the energy range E$_x $=(20$-$40)MeV in $^{12}$C are quenched by about 20 \% due to the coupling to the 3$\hbar \omega $ configuration space, and by additional 10\% due to the tensor interaction. There is also a similar quenching of spin-dipole strengths for 1$^- $ states in $^{16}$O. The calculated results are compared with available experimental data of spin-dipole states observed by charge exchange $(p,n)$ and $(\vec{d}, ^{2}$He) reactions on $^{12}$C. An astrophysical implication of spin-dipole strengths in $^{16}$O for detections of supernova neutrinos is also discussed.
An exclusive coincidence experiment of the $(p,n_{\rm IAS} \tilde{p})$ reaction was carried out for the first time by measuring decay protons from the isobaric analog states (IAS) formed by the $(p,n)$ reactions on $^{140}$Ce, $^{172,174,176}$Yb, and $^{208}$Pb at $E_p $= 35 MeV. Spreading widths of the IAS were deduced from the escape and total widths. Results were compared with a recent theory of isospin mixing, corroborating the conclusion that contributions from the isovector monopole state, especially that from the $T$-component instead of that from the $T_0 -1 $ component, were important to explain quantitatively the experimental $A$ -dependence of spreading widths. $A$ and $Z$ dependence of isospin mixing was determined to be $(7.8 \pm 0.6) \times 10^{-7} A^{2/3} Z^2 $ by comparing the results with the theory.
We study displacement fields of multipole excitations in a halo nucleus $^{11}_4 $Be$_{7}$ and a skin nucleus $^{60}_{20}$Ca$_{40}$~ in comparison with those of collective states in a $\beta-$stable nucleus $^{208}_{82}$Pb$_{126}$. The self-consistent Hartree-Fock calculation plus the random phase approximation with Skyrme interactions is performed in the coordinate space, taking into account the proton and neutron degree of freedom. Transition densities, radial current components and displacement fields of giant resonances in drip line nuclei are found very similar to those in $\beta$ stable nuclei. The isovector giant resonances are not properly understood in terms of any available collective models, while the isoscalar giant resonances are in a good approximation expressed by collective models. It is found that the radial velocity of the highest-lying peak of isovector giant multipole resonances vanishes at a radius slightly outside of the surface. It is shown that transition densities and current components of the excitations just above the threshold of the drip line nuclei are extended to far outside of the nuclear surface, where the displacement fields show a simple pattern.
Explicit relations between the isospin mixing probability, the spreading width $\Gamma_{IAS}^{\downarrow}$ of the Isobaric Analog State (IAS) and the statistical decay width $\Gamma_c$ of the compound nucleus % with $J^{\pi}$=0$^+$ at finite excitation energy, are derived by using the Feshbach projection formalism. The temperature dependence of the isospin mixing probability is discussed quantitatively for the first time by using the values of $\Gamma_{IAS}^{\downarrow}$ and of $\Gamma_c$ calculated by means of microscopic models. It is shown that the mixing probability remains essentially constant up to a temperature of the order of 1 MeV and then decreases to about 1/4 of its zero temperature value, at higher temperature than $\approx$ 3 MeV, due to the short decay time of the compound system.
The incompressibility of asymmetric nuclear matter $K_{\in fty}$ is studied analytically within the framework of the relativistic mean field theory and the non-relativistic Skyrme Hartree Fock model. We investigate also the relation between $K_{\infty}$ of asymmetric nuclear matter and the surface diffuseness by using the extended Thomas Fermi approximation. The self-consistent relativistic and non-relativistic mean field calculations are performed for Sn isotopes, taking into account the pairing correlation, in order to extract the asymmetric parameter $(N-Z)/A$ dependences of the surface diffuseness $a$ and the central density $\rho(r=0)$. A clear correlation is found between the incompressibility $K_{\infty}$ of asymmetric nuclear matter and the extracted surface diffuseness $a$, and also between $K_{\infty}$ and the extracted central density $\rho(r=0)$ .
We have studied dipole states of oxygen-isotopes in large scale shell model calculations. The calculated photoreaction cross sections in $^{16}$O and $^{18}$O give reasonable agreement with experimental observations both in the low energy region below $\hbar \omega $=15 MeV and in the high energy giant resonance region (15 MeV $<\hbar \omega \leq $30 MeV). We found that the transition strength below dipole giant resonance ($\hbar \omega \leq $15 MeV) exhausts about 10\% of the classical Thomas-Reiche-Kuhn sum rule value in heavier oxygen isotopes than $^{18}$O. The T$_>$ GDR appears in $^{18}$O and $^{20}$O having comparable transition strengths with the T$_<$ GDR, while T$_>$ strengths become much smaller than T$_<$ ones in $^{22}$O and $^{24}$O.
The nuclear magnetic-dipole moment and electric quadrupole moment of the short-lived $\beta $-emitting nucleus $^{19}$O have been determined by detecting its $\beta $-NMR in CaO and TiO$_2$ crystals. Slight deviations of both values from the shell model predictions given by Hartree-Fock calculations demand further experimental and theoretical studies.
Triple systems and Lie algebras.
On isotopies and octonions.
A generalization of structurable algebras.
On Jordan-Lie triple systems.
Triple systems and Yang-Baxter equation.
An Applications of Lie algebras.
Wedderburn theorem of triple systems.
In this paper, Bayes decision theory is combined with the approximation theory on three-layer neural networks, and the two-category n-dimensional Gaussian classification problem is studied. First, we prove theoretically that three-layer neural networks with at least 2n hidden units have the capability of approximating the a posteriori probability in the two-category classification problem with arbitrary accuracy. Second, we prove that the input-output function of neural networks with at least 2n hidden units tends to the a posteriory probability as Back-Propagation learning ideally.
A boolean tableau is an array $T=(T_{ij})$ of the elements of a finite boolean algebra with several rows and infinitely many columns, where the entries increase from left to right and downwards. We study the generating functions for various classes of boolean tableaux. Applying the Gessel-Viennot method to certain non-planar digraphs, we have determinantal formulas for the generating functions, which are regarded as generalized Jacobi-Trudi identities. By this theorem, we can also deal with {\it ideal-tableaux of zigzags}, and give some new totally positive matrices.
We define a natural generalization of order ideals, ideals of a digraph $D$. T he ideal-generating function $f(D,p)$ of $D$ is defined as $\sum_I\prod_xp_x$, where the sum runs over all ideals of $D$, and the product all vertices of $I $. Let $\D=\{D_1,\dots,D_s\}$ be a set of induced subdigraphs of $D$ such that $\coprod V(D_i)=V(D)$ (section decomposition), where $V(D)$ denotes the set o f vertices of $D$. Denote by $s(D/\D)$ the totality of the digraphs obtained f rom $D$ by reversing directions of some edges in $E(D)-E(D_1)-\dots-E(D_s)$, w here $E(D)$ denotes the set of edges of $D$. We prove the formula: $f(D,p)=(1+\w)^{-n}\sum_F\w^{\r(D,F)}\prod_{i=1}^sf(D_i,(\w^{m(F;x)}p_x)_x)$. Here, $F$ ranges over $s(D/\D)$, $m(F;x)=\#\{e\in E(F)-E(D_1)-\dots-E(D_s);e\ \text{is directed to}\ x\}-\#\{\t ext{such edges from $x$}\}$, $\r(D,F)$ denotes the number of edges of $F$ whos e directions are opposite to the corresponding edges of $D$, and $\w=$primitiv e 6th root of 1. In virtue of this, we can compute the ideal-generating funct ions in one variable $q$ of cyclically linked copies of a symmetric digraph.
The closed shell structure at N=Z=28 is studied by a large-scale shell model calculation by the quantum Monte Carlo diagonalization method. Latest crucial improvements of the method are described. The doubly closed shell probability of 56Ni is shown to be only 49\% in a full pf shell calculation, in contrast to the corresponding probability of 48Ca which reaches 86\%.
We introduce the Bessel transforms of increasing Levy processes and prove that increasing Levy processes with the distributions at time 1 being n-mixtures of exponential distributions are at most n-modal processes. Moreover we give a full description of modality of the Bessel transforms of increasing self-decomposable processes.
Absolute continuity and smoothness of distributions in the nested subclasses of the class of all B-decomposable distributions are studied. All invertible matrices are classified into two typ es in terms of P.V. numbers. The minimum integer m for which all full distribu tions in the m-th subclass are absolutely continuous and the minimum integer m for which all absolutely continuous distributions in the m-th subclass have t he r-fold differential densities are discussed according to the type of the m atrix B related to P.V. numbers.
There are two types of generalizations of selfdecomposablity of probability measures : the c-decomposability and the C- decomposability of Loeve and Bunge on the one hand, and the semi-selfdecomposa bility of Maejima-Naito on the other. The latter implies infinite divisibility but the former does not in general. Intr oduction of operator normalizations yields four kinds of classes of distributi ons. Further, each of these classes generates the Urbanik-Sato type decreasing sequence of its subclasses. Characterizations and relations of theses classes and subclasses are established.
The purpose of this paper is to show that a generating function of the Jacobi polynomials can be regarded as the integral kernel of a unitary mapping from an $L^{2}$ space onto a Hilbert space of analytic functions.
Refereed Proceeding Papers
A Microscopic Study of Giant Resonances.
$\beta -$Stable and Drip Line Nuclei.
Jordan superalgebras and Lie superalgebras.
Recent developments of the Quantum Monte Carlo Diagonalization method for large scale shell model calculations are presented. By introducing a quasi- variational aspect into the stochastic basis-sampling process, the number of basis states can be reduced drastically. As a result the exact symmetry restoration becomes possible. The accuracy of realistic calculations in the pf-shell is examined for 48Cr. As an application the wave function of 56Ni is analyzed, which shows significant breaking of doubly magic closed shell.
Academic Activities
Others