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Hitoshi Nakada

Publications and source records attributed to Hitoshi Nakada.

18 recordsLinked to original sources
arXiv2026

Farey graphs and geodesic expansions of complex continued fractions

We discuss complex Farey graphs for the Euclidean imaginary quadratic number fields $\mathbb Q(\sqrt{-d})$, $d\in\{1, 2, 3, 7, 11\}$. We study hyperbolic versions of A. Schmidt's Farey polygons living in $3$-dimensional hyperbolic space $\mathbb{H}^3$. Using these Farey polygons we recover tessellations of the hyperbolic plane $\mathbb{H}^2$ that are defined by the action of the Hecke groups $H_4$ and $H_6$ and have been studied earlier by I. Short and M. Walker. Moreover, hyperbolic Farey polygons allow us to define polyhedra that induce Farey tessellations of $\mathbb{H}^3$ by the action of certain Bianchi groups. Using complex Farey graphs we consider geodesic complex continued fraction expansions. Our method provides a different and more general approach as the one from the discussion by M. Hockman.

math.NT↗
arXiv2024

Extravagance, irrationality and Diophantine approximation

For an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotient function is integrable. We show that with respect to a ``continued fraction mixing'' measure for the Gauss map with the log of the partial quotient function non-integrable, almost all numbers are Liouville. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are applications of our study of the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type dichotomy for Diophantine approximation with respect to ergodic``weak Renyi measures'' which are ``doubling at $0$''.

math.DS↗
arXiv2024

A new class of $\alpha$-Farey maps and an application to normal numbers

We define two types of the $\alpha$-Farey maps $F_{\alpha}$ and $F_{\alpha, \flat}$ for $0 < \alpha < \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le \alpha \le 1$ by R.~Natsui (2004). Then, for each $0 < \alpha < \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{\alpha, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associted with $\alpha$-continued fractions does not vary by the choice of $\alpha$, $0 < \alpha < 1$. This extends the result by C.~Kraaikamp and H.~Nakada (2000).

math.DS↗
arXiv2017

Discrepancy Skew Products and Affine Random Walks

We prove bounded rational ergodicity for some discrepancy skew products whose rotation number has bad rational approximation. This is done by considering the asymptotics of associated affine random walks.

math.DS↗
arXiv2014

Quantum criticality in an asymmetric three-leg spin tube: A strong rung-coupling perspective

We study quantum phase transitions in the asymmetric variation of the three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one of the rung exchange couplings $J'_\perp$ while the other two are kept constant $J_\perp$. We focus on the strong rung-coupling regime $J_\perp \gg J_\parallel$, where $J_\parallel$ is the leg coupling, and analyze the effective spin-orbital model with a transverse crystal field in detail. Applying the Abelian bosonization to the effective model, we find that the system is in the dimer phase for the general half-odd-integer-spin cases without the rung modulation; the phase transition between the dimer and Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a level spectroscopy analysis for the effective model for spin-1/2 using exact diagonalization, to determine the precise transition point $| J'_\perp - J_\perp| /J_\parallel \sim 0.283$ in the strong rung-coupling limit. The presence of the dimer phase in a small but finite region is also confirmed by a density-matrix renormalization group calculation on the original spin-tube model.

cond-mat.str-el↗
arXiv2014

Farey map, Diophantine approximation and Bruhat-Tits tree

Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.

math.DS↗
arXiv2012

Nuclear Level Density of ${}^{161}$Dy in the Shell Model Monte Carlo Method

We extend the shell-model Monte Carlo applications to the rare-earth region to include the odd-even nucleus ${}^{161}$Dy. The projection on an odd number of particles leads to a sign problem at low temperatures making it impractical to extract the ground-state energy in direct calculations. We use level counting data at low energies and neutron resonance data to extract the shell model ground-state energy to good precision. We then calculate the level density of ${}^{161}$Dy and find it in very good agreement with the level density extracted from experimental data.

nucl-th↗
arXiv2009

Non-local mean field effect on nuclei near Z=64 sub-shell

Evolutions of single-particle energies and Z=64 sub-shell along the isotonic chain of N=82 are investigated in the density dependent relativistic Hartree-Fock (DDRHF) theory in comparison with other commonly used mean field models such as Skyrme HF, Gogny HFB and density dependent relativistic Hartree model (DDRMF). The pairing is treated in the BCS scheme, except for Gogny HFB. It is pointed out that DDRHF reproduces well characteristic features of experimental $Z$-dependence of both spin orbital and pseudo-spin orbital splittings around the sub-shell closure Z=64. Non-local exchange terms of the isoscalar $σ$ and $ω$ couplings play dominant roles in the enhancements of the spin-orbit splitting of proton 2d states, which is the key ingredient to give the Z=64 sub-shell closure properly. On the other hand, the $π$ and $ρ$ tensor contributions for the spin-orbit splitting cancel each other and the net effect becomes rather small. The enhancement of the sub-shell gaps towards Z=64 is studied by the DDRHF, for which the local terms of the scalar and vector meson couplings are found to be important.

nucl-th↗
arXiv2009

138La-138Ce-136Ce nuclear cosmochronometer of supernova neutrino process

The 138La (T1/2=102 Gyr) - 138Ce - 136Ce system is proposed to be used as a nuclear cosmochronometer for measuring the time elapsed from a supernova neutrino process. This chronometer is applied to examine a sample affected by a single nucleosynthesis episode as presolar grains in primitive meteorites. A feature of this chronometer is to evaluate the initial abundance ratio of 136Ce/138Ce using an empirical scaling law, which was found in the solar abundances. We calculate the age of the sample as a function of isotopic ratios, 136Ce/138Ce, and 138La/138Ce, and evaluate the age uncertainty due to theoretical and observational errors. It is concluded that this chronometer can work well for a sample with the abundance ratio of 138La/138Ce >= 20 when the ratios of 136Ce/138Ce and 138La/138Ce are measured within the uncertainty of 20%. The availability of such samples becomes clear in recent studies of the presolar grains. We also discuss the effect of the nuclear structure to the neutrino process origin of 138La.

astro-ph.SR↗
arXiv2008

Metric and arithmetic properties of mediant-Rosen maps

A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.

math.NT↗
arXiv2007

On the Lenstra constant associated to the Rosen continued fractions

The purpose of this paper is to describe the relation between the Legendre and the Lenstra constants. Indeed we show that they are equal whenever the Legendre constant exists; in particular, this holds for both Rosen continued fractions and $α$-continued fractions. We also give the explicit value of the entropy of the Rosen map with respect to the absolutely continuous invariant probability measure.

math.NT↗
arXiv2007

Generalized Brjuno functions associated to $α$-continued fractions

For αin the interval [0,1], we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α=1) and the nearest integer (α=1/2) and by-excess (α=0) continued fraction maps. To each of these expansions, and to each choice of a positive function u on the interval I_α=(0,max(α,1-α)) we associate a generalized Brjuno function B_(α,u)(x). For α=1/2 or α=1, and u(x)=-\log(x), these functions were introduced by Yoccoz in his work on the linearization of holomorphic maps. Their regularity properties, including BMO regularity and their extension to the complex plane, have been thoroughly investigated. We compare the functions obtained with different values of αand we prove that the set of (α,u)-Brjuno numbers does not depend on the choice of αprovided that α>0. We then consider the case α=0, u(x)=-\log(x) and we prove that x is a Brjuno number (for α> 0) if and only if both x and -x are Brjuno numbers for α=0.

math.DS↗
arXiv2006

On the mixing coefficients of piecewise monotonic maps

We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse $ϕ$-mixing. If its invariant density is uniformly bounded away from 0, it is $ϕ$-mixing iff all images of all orders are big in which case it is $ψ$-mixing. Among $\b$-transformations, non-$ϕ$-mixing is generic. In this sense, the asymmetry of $ϕ$-mixing is natural.

math.DS↗
arXiv2006

Exchangeable, Gibbs and equilibrium measures for Markov subshifts

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of "symmetric measure": exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.

math.PR↗
arXiv1993

E2 properties of nuclei far from stability and the proton-halo problem of 8B

E2 properties of A=6--10 nuclei, including those of nuclei far from stability, are studied by a $(0+2)\hbarω$ shell-model calculation which includes E2 core-polarization effects explicitly. The quadrupole moments and the E2 transition strengths in A=6--10 nuclei are described quite well by the present calculation. This result indicates that the relatively large value of the quadrupole moment of $^8$B can be understood without introducing the proton-halo in $^8$B. An interesting effect of the $2\hbarω$ core-polarization is found for effective charges used in the $0\hbarω$ shell model; although isoscalar effective-charges are almost constant as a function of nucleus, appreciable variations are needed for isovector effective-charges which play important roles in nuclei with high isospin-values.

nucl-th↗
arXiv1993

Microscopic description of nuclei in the middle of the pf-shell by a shell model calculation with G-matrix interaction

Energy levels and electromagnetic properties of with $N=28\sim 30$ nuclides are studied in terms of a large-scale shell model calculation, which contains no newly adjusted parameters. The Kuo-Brown $G$-matrix interaction is shown to reproduce energy levels of 205 low-lying states of these nuclei. We evaluate effective charges by incorporating the core-polarization effects caused by the coupling to GQR's. We then compute E2 moments and transition probabilities. The M1 moments and transition rates are calculated by quoting the effective $g$-factors of Towner, which are obtained by taking into account the meson-exchange and the core-polarization mechanisms. By this microscopic calculation most of the E2 properties and the magnetic moments are reproduced. Although there are agreements and disagreements in the M1 transition rates, the general tendency is reproduced. The $(e,e')$ and $(p,p')$ excitation from the ground state to some low-lying $2^+$ states is also discussed.

nucl-th↗