/ Noriaki Kamiya / Professor
/ Hiroyuki Sagawa / 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 research and education 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: non-associative algebras, neural networks, unimodality problems, algebraic combinatorics, spherical functions, and homotopy theory.
In the fields of physics, theoretical research is performed in many-body theories, nuclear physics , quantum gravity and quantum mechanics of constrait system. 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
We have studied spin-orbit interactions in Skyrme Hartree-Fock and relativistic mean field theories. We pointed out that the two models give different results not only in the spin-orbit splitting but also in the $\bf{l.l}$ splitting. The single particle energies of $^{72}$Kr is studied focusing on the 2p-1f shell.
We have studied electric dipole (E1) and spin-dipole strength distribution in $^{11}$Li by shell model calculations with halo effect. Two peaks in the E1 strength are found in the low excitation energy region below E$_{x}$ = 4 MeV, which have almost the same energies as observed E1 peaks in the Coulomb breakup reaction of $^{11}$Li. The calculated E1 strength up to E$_{x}$ = 4 MeV exhaust about 4 $\%$ of the Thomas-Reiche-Kuhn energy-weighted sum rule value. We found also pigmy and giant peaks in spin-dipole strengths of $^{11}$Li. Possible existence of double soft dipole mode and giant resonance built on the soft dipole states are investigated.
Using the self-consistent Hartree-Fock plus Tamm-Dancoff approximation with Skyrme interactions, the response of some $\beta$ stable closed-shell nuclei to charge-exchange (t$_{\pm}$) spin monopole fields is estimated in the coordinate space. Both the isoscalar and isovector spin-correlation are included simultaneously. The excitation energy of t$_+$ giant resonance decreases considerably as the neutron excess (and consequently the nuclear mass number) increases, while that of t$_-$ giant resonance remains nearly constant. We have tried to separate out the Gamow-Teller strength from the present charge-exchange spin monopole strength, both of which have the same quantum numbers, $J^{\pi}$=1$^+$.
Gamow-Teller (GT) and spin-dipole (SD) states in $^{208}$Bi are studied by using self-consistent Hartree-Fock + Tamm-Dancoff approximation taking into account the coupling to the continuum. We study also electric dipole (E1) transitions between GT and SD states in $^{208}$Bi. Most of SD strength is found at the excitation energy E$_x \approx $25MeV with a very board width. The Landau damping effect is responsible for the large width of SD peak, while the escape width is found at most 1MeV. Main E1 transitions for 0$^{-}$ and 1$^{-}$ states are found near excitation energy expected from Brink's hypothesis in which SD states are considered as E1 resonances built on the GT state. For 2$^{-}$, a half of the E1 strength is fragmented among many state in lower energy region below the expected energy. Calculated E1 transition strengths between GT and SD states are compared with the analytic sum rules within one-particle one-hole (1p-1h) configuration space and within both 1p-1h and 2p-2h model space.
Inelastic scattering of the neutron-rich nucleus $^{12}$Be on lead and carbon targets has been studied by measuring de-excitation $\gamma$ rays in coincidence. The strong $\gamma$-ray transition from the state at $E_{x}$~$=$~2.68(3)~MeV following E1 Coulomb excitation was observed for the lead target, leading to an assignment of $J^{\pi}$~=~1$^{-}$ for the excited state. The low excitation energy of this intruder 1$^{-}$ state and the deduced large $B(\rm{E1};0_{g.s.}^{+}\to1^{-})$ value of 0.051(13)~e$^{2}$fm$^{2}$ provide a consistent picture of the $N$$=$8 shell melting in $^{12}$Be.
We propose a prescription to improve spin-orbit potentials of Skyrme interaction in N$>$Z nuclei. New spin-orbit parameters for Skyrme interactions SIII, SkM$^{*}$ and SGII are shown to give better agreement with empirical spin-orbit splittings than the original ones not only in $^{16}$O but also in $^{208}$Pb. The isotope shifts in Pb nuclei and excitation energies of collective Gamow-Teller states in $^{48}$Sc, $^{90}$Nb and $^{208}$Bi are calculated successfully by using new spin-orbit parameters.
We have studied electric dipole (E1) strength distribution in neutron and proton drip line nuclei $^{12}_{\,\,\,4}$Be$_8$, $^{14}_{\,\,\,4}$Be$_{10}$ and $^{13}_{\,\,\,8}$O$_5$ by large scale shell model calculations with the effect of loosely-bound single particle wave functions. Large E1 strength are found in the low excitation energy region below E$_{x}$=3MeV in both $^{12}_{\,\,\,4}$Be$_8$ and $^{13}_{\,\,\,8}$O$_5$, while strong E1 strength are found only in high excitation energy region above E$_x$=7MeV in $^{14}_{\,\,\,4}$Be$_{10}$. The effect of extended wave functions and the coherence in the transition amplitudes enhance significantly the E1 strength in the low energy region. A manifestation of melting of the shell magicity at $N$=8 and $Z$=8 is pointed out in the systematics of E1 transitions in Be- and O-isotopes.
Let $F$ be a monoid of countably many functions holomorphic at $y^0$, and $(x_f)_{f\in F}$ be a set of independent variables. We set $F_*=F-\{1\}$, $x_*=(x_f)_{f\in F_*}$. Let $(F_1,F_2,\dots)$ be an increasing sequence of finite subsets of $F$ such that $\bigcup_{i\ge1}F_i=F$. For $i\ge1$, let $A_i$ denote the ring of all functions of $(x_f)_{f\in F_i}$, holomorphic at $(x_1,(x_f)_{f\in F_i-\{1\}})=(x_1^0,0)$. Define $A=\projlim A_i$. Consider the implicit function $y\in A$ defined by $g(y)=\sum_{f\in F}x_ff(y)$ $(y(x_1^0,0)=y^0)$. We have the Taylor expansion of $y$ at $x_*=0$: $y=g^{-1}(x_1)+\sum_{\alpha}\Bigl({d^{|\alpha|-1}\over dx_1^{ |\alpha|-1}} {\prod_{f\in F_*}f^{\alpha(f)}(g^{-1}(x_1)) \over g'(g^{-1}(x_1))}\Bigr) {x_*^{\alpha}\over\alpha!}$, where the sum runs over all maps $\alpha:F_*\longrightarrow\{0,1,2\dots\}$ such that $|\alpha| :=\sum_{f\in F_*}\alpha(f)$ are positive finite.
In this paper we deal with a generating function of zonal spherical functions on $U(n)/U(n-1)$ and its application to characterization of the zonal spherical functions.
A spherical-deformed shape coexistence for semimagic $pf$-shell nuclei is investigated. Deformed states can arise by breaking $f_{7/2}$ submagic shell, and coexist with spherical yrast states for $^{54}$Fe. These states are shown to be primarily two-particle two-hole excitations on top of the respective correlated ground states. In addition to these modes, a four-particle-four-hole excitation can be found for $^{54}$Fe, which is related to the deformed band of doubly magic $^{56}$Ni. To investigate this shape coexistence, we use the conventional shell-model diagonalization, constrained Hartree-Fock, generator coordinate, variation- after-projection, and Monte Carlo shell-model methods.
The transition from spherical to deformed shapes is studied in terms of large-scale shell-model calculations for Ba isotopes as a function of valence nucleon number with fixed single-particle space and Hamiltonian. A new version of the Monte Carlo shell model is introduced so as to incorporate pairing correlations efficiently, by utilizing condensed pair bases. The energy levels and electromagnetic matrix elements are described in agreement with experiments throughout the transitional region. The orbital $M1$ sum rule is calculated as a measure of the deformation evolution, and $Q$-phonon picture is shown to be reasonable from spherical to deformed nuclei.
Refereed Proceeding Papers
I report on several new results of pigmy and giant resonances in nuclei near drip lines. Soft dipole states in halo nuclei will be also discussed. Main topics are as follows;
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 about 20 \% by the c oupling to the 3$\hbar \omega $ configuration space, and additional 10\% by the tensor interaction. There is also a similar quenching of spin-dipole strength s for 1$^- $ states in $^{16}$O. The calculated results in $^{12}$C 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.
Electric dipole states in light neutron-rich nuclei are studied by large scale shell model calculations. The calculated photoreaction cross sections in oxygen-isotopes and $^{14}$C 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, which is consistent with new experimental data from GSI. Isospin structure of GDR is also studied projecting out the isospin in the shell model basis.
We have studied dipole states of oxygen-isotopes in large scale shell model calculations. The calculated photoreaction cross sections in $^{16}$O, $^{17}$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 $15MeV) exhausts about 10\% of the classical Thomas-Reiche-Kuhn sum rule and more than 40\% of the cluster sum rule in heavier oxygen isotopes than $^{17}$O. The predicted Pigmy strengths in $^{20}$O and $^{22}$O are also confirmed by recent Coulomb excitation experiment at GSI.
Let $O$ be a finite odd-ary tree as a digraph or equivalently, as an ordered set. Let $K=(K_1,\dots,K_r)$ be connected induced subdigraphs of $O$, including all the edges incident with ramification vertices of $O$. We deal with tableaux of order ideals of $K_1,\dots,K_r$, with $r$ rows and infinitely many columns. The $(i,j)$-entry $T_{ij}$ of the tableau is an order ideal of $K_i$, which increases as $j$ increases, and satisfies $T_{ij}\cap K_{i+1+l}\sub set T_{i+1+l,j+l}\cap K_i$. Let the map $B_i$ from end vertices, say, vertices of degree $\le1$ of $K_i$, to $\bZ$ be fixed. Also take a map $\alpha$ from edges of $O$ to intervals of non negative integers. We define a certain class ${\rm Tab}(K,B,\alpha)$ of the above tableaux and construct the bijection between that class and a set of ``tree-like'' disjoint paths in a special digraph. In virtue of this, together with an odd-ary-tree-analogue of Lindstr\"om's theorem, we prove the Jacobi--Trudi identity for ${\rm Tab}(K,B,\alpha)$ with a tensor determinant of even dimension $s=\#\{{\rm end vertices of }O\}$.
Books
Temporal changes in modality and absolute continuity of the distributions of Levy processes are studied. Examples of Levy processes whose distributions can drastic temporal changes from unimodal to multimodal and from continuous and singular to absolutely continuous are given.
Academic Activities
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