An inverse problem of the flux for minimal surfaces

For a complete minimal surface in the Euclidean 3-space, the so-called flux vector corresponds to each end. The flux vectors are balanced, i.e., the sum of those over all ends are zero. Consider the following inverse problem: For each balanced n vectors, find an n-end catenoid which attains given vectors as flux. Here, an n-end catenoid is a complete minimal surface of genus 0 with ends asymptotic to the catenoids. In this paper, the problem is reduced to solving algebraic equation. Using this reduction, it is shown that, when n=4, the inverse problem for 4-end catenoid has solutions for almost all balanced 4 vectors. Further obstructions for n-end catenoids with parallel flux vectors are also discussed.Comment: 28 pages, AMSLaTeX 1.1, with 8 figures, To appear in Indiana University Mathematics Journa

Chiral Mass Splitting for c \bar{s} and c \bar{n} Mesons in the \tilde{U}(12)-Classification Scheme of Hadrons

We investigate the chiral mass splitting of parity-doubled J=0,1 states for c \bar{s} and c \bar{n} meson systems in the \tilde{U}(12)_{SF}-classification scheme of hadrons, using the linear sigma model to describe the light-quark pseudoscalar and scalar mesons together with the spontaneous breaking of chiral symmetry, and consequently predict the masses of as-yet-unobserved (0^{+},1^{+}) c \bar{n} mesons. We also mention some indications of their existence in the recent published data from the Belle and BABAR Collaborations.Comment: 5 pages, 1 figures, talk at the 11th International Conference on Hadron Spectroscopy, Rio de Janeiro, Brazil, 21-26 Aug 2005, to appear in the Proceeding

Random Wandering Around Homoclinic-like Manifolds in Symplectic Map Chain

We present a method to construct a symplecticity preserving renormalization group map of a chain of weakly nonlinear symplectic maps and obtain a general reduced symplectic map describing its long-time behaviour. It is found that the modulational instability in the reduced map triggers random wandering of orbits around some homoclinic-like manifolds, which is understood as the Bernoulli shifts.Comment: submitted to Prog. Theor. Phy

Phase structure of NJL model with weak renormalization group

We analyze the chiral phase structure of the Nambu--Jona-Lasinio model at finite temperature and density by using the functional renormalization group (FRG). The renormalization group (RG) equation for the fermionic effective potential $V(\sigma;t)$ is given as a partial differential equation, where $\sigma:=\bar \psi\psi$ and $t$ is a dimensionless RG scale. When the dynamical chiral symmetry breaking (D$\chi$SB) occurs at a certain scale $t_c$, $V(\sigma;t)$ has singularities originated from the phase transitions, and then one cannot follow RG flows after $t_c$. In this study, we introduce the weak solution method to the RG equation in order to follow the RG flows after the D$\chi$SB and to evaluate the dynamical mass and the chiral condensate in low energy scales. It is shown that the weak solution of the RG equation correctly captures vacuum structures and critical phenomena within the pure fermionic system. We show the chiral phase diagram on temperature, chemical potential and the four-Fermi coupling constant.Comment: 32 pages, 12 figures; Version published in Nuclear Physics