The digit exchanges in the beta expansion of algebraic numbers

In this article, we investigate the $\beta$-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where $\beta$ is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers. Our method for the number of digit exchanges is also applicable to more general representations of complex algebraic numbers $\xi$ by infinite series $\xi=\sum_{n=1}^{\infty} t_n \beta^{-n}$, where $\bold{t}=(t_n)_{n\ge 1}\in \Z^{\N}$ is a bounded sequence of integers and $\beta$ is a quasi-Pisot or quasi-Salem number.Comment: 12 page

Endoscopy-guided vitreoretinal surgery following penetrating corneal injury: a case report

Motoko Kawashima1, Shinichi Kawashima2, Murat Dogru1,3, Makoto Inoue4, Jun Shimazaki1,51Department of Ophthalmology, Tokyo Dental College, Chiba, Japan; 2Department of Ophthalmology, International University of Health and Welfare, Tokyo, Japan; 3Department of Ocular Surface and Visual Optics, Keio University School of Medicine, Tokyo, Japan; 4Kyorin Eye Center, Tokyo, Japan; 5Department of Ophthalmology, Keio University School of Medicine, Tokyo, JapanIntroduction: Severe ocular trauma requires emergency surgery, and a fresh corneal graft may not always be available. We describe a case of perforating eye injury with corneal ­opacity, suspected endophthalmitis, and an intraocular foreign body. The patient was successfully treated with a two-step procedure comprising endoscopy-guided vitrectomy followed by corneal transplantation. This surgical technique offers a good option to vitrectomy with simultaneous keratoplasty in emergency cases where no graft is immediately available and there is the ­possibility of infection due to the presence of a foreign body.Case presentation: A 55-year-old Japanese woman was referred to our hospital with a ­perforating corneal and lens injury sustained with a muddy ferrous rod. Primary corneal sutures and lensectomy were performed immediately. Vitreoretinal surgery was required due to ­suspected endophthalmitis, vitreous hemorrhage, retinal detachment, dialysis and necrosis of the peripheral retina. Instead of conventional vitrectomy, endoscopy-guided vitreous surgery was performed with the Solid Fiber Catheter AS-611 (FiberTech, Tokyo, Japan) due to the presence of corneal opacity and the unavailability of a donor cornea. The retina was successfully attached with the aid of a silicon oil tamponade. Following removal of the silicon oil at 3 months after surgery, penetrating keratoplasty and intraocular lens implantation with ciliary sulcus suture fixation were performed. At 6 months after penetrating keratoplasty, the graft remained clear and visual acuity was 20/40.Conclusion: Primary endoscopic surgery for vitreoretinal complications in eyes with perforating injury performed prior to penetrating keratoplasty appears to be advantageous in terms of avoiding damage to the corneal endothelium.Keywords: vitreoretinal surgery, emergency, foreign bod

Can polylogarithms at algebraic points be linearly independent?

Let $r,m$ be positive integers. Let $0\le x <1$ be a rational number. Let $\Phi_s(x,z)$ be the $s$-th Lerch function $\sum_{k=0}^{\infty}\tfrac{z^{k+1}}{(k+x+1)^s}$ with $s=1,2,\ldots ,r$. When $x=0$, this is the polylogarithmic function. Let $\alpha_1,\ldots ,\alpha_m$ be pairwise distinct algebraic numbers with $0<|\alpha_j|<1$ $(1 \le j \le m)$. In this article, we state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers $:$ $\Phi_1(x,\alpha_1),\Phi_2(x,\alpha_1),\ldots, \Phi_r(x,\alpha_1),\Phi_1(x,\alpha_2),\Phi_2(x,\alpha_2),\ldots, \Phi_r(x,\alpha_2),\ldots,\Phi_1(x,\alpha_m),\Phi_2(x,\alpha_m),\ldots, \Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the $r$ Lerch functions $\Phi_1(x,z),\Phi_2(x,z),\ldots, \Phi_r(x,z)$ at $m$ distinct algebraic points without any assumption for $r$ and $m$, even for the case $x=0$, the polylogarithms. We give an outline of our proof and explain basic idea

SS-unit equation in two variables and Pad\'{e} approximations

In this article, we use Pad\'{e} approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Pad\'{e} approximants with a simple argument relying on Mahler measure and on the local height, we refine the bound due to J.-H. Evertse.Comment: 13 page

Glycemic control in type 2 diabetes

Type 2 diabetes is a typical lifestyle disease. We aimed to identify the factors affecting glycemic control in 64 outpatients with type 2 diabetes over a 2-year period. We defined poor glycemic control using a change in glycosylated hemoglobin (ΔHbA1c) of ≥ 0.5% over 2 years and/or HbA1c ≥ 7.5% at the end of the study period. We used a questionnaire to collect information on oral health behavior and lifestyle, including eating and smoking habits, and analyzed the relationships between indices of diabetes control and responses to the questionnaire. The mean (SD) HbA1c of the participants was 6.87% (0.77%) at a baseline, and 6.93% (0.69%) after 2 years. Twenty-three participants (36.0%) had poor glycemic control. ΔHbA1c and the change in body mass index (ΔBMI) correlated (Spearman’s rank correlation, r = 0.350, p < 0.01). The HbA1c at baseline was associated with eating slowly / chewing well, and ΔBMI was associated with perceived oral symptoms. Binominal logistic regression analysis revealed that poor glycemic control was associated with ΔBMI and a smoking habit (odds ratio : 1.62, 95% confidence interval : 1.08–2.42 ; and 4.01, 1.12–14.36, respectively). These findings imply that weight gain and a smoking habit are associated with poor glycemic control in patients with type 2 diabetes

Diabetes indicators and oral health behavior

Objective : The aim of this study was to evaluate the association between diabetes-related indicators and oral health behavior among patients with type 2 diabetes. Methods : Seventy-four outpatients were included. We administered a questionnaire and divided the patients into two groups according to oral health behavior and eating habits. We then compared diabetes-related clinical records between the groups and performed logistic regression analysis. Results : Participants who brushed their teeth before bedtime every night had lower BMIs than those who did not. Participants who reported eating slowly and chewing well every day had significantly lower HbA1c than their counterparts. Participants who reported gum bleeding had significantly higher LDL-cholesterol than their counterparts. Binominal logistic regression analysis revealed that BMI < 25 was associated with not brushing teeth before bedtime every night, HbA1c < 7.5 with not eating slowly or chewing well every day, and LDL-cholesterol < 120 with gum bleeding (odds ratio : 0.140, 95% confidence interval : 0.036–0.540 ; OR : 0.085, 95% CI : 0.010.0–736, OR : 0.275, 95% CI : 0.077.0–979, respectively). Conclusions : Our findings suggest that toothbrushing before bedtime every night is associated with reduced risk of obesity and that eating slowly and chewing well are advantageous for glycemic control in patients with type 2 diabetes

Chemical effect on muonic atom formation through muon transfer reaction in benzene and cyclohexane samples

To investigate the chemical effect on the muon capture process through a muon transfer reaction from a muonic hydrogen atom, the formation rate of muonic carbon atoms is measured for benzene and cyclohexane molecules in liquid samples. The muon transfer rate to carbon atoms of the benzene molecule is higher than that to the carbon atoms of the cyclohexane molecule. Such a deviation has never been observed among those molecules for gas samples. This may be because the transfers occur from the excited states of muonic hydrogen atoms in the liquid system, whereas in the gas system, all the transfers occur from the 1s (ground) state of muon hydrogen atoms. The muonic hydrogen atoms in the excited states have a larger radius than those in the 1s state and are therefore considered to be affected by the steric hindrance of the molecular structure. This indicates that the excited states of muonic hydrogen atoms contribute significantly to the chemical effects on the muon transfer reaction

学生による授業評価は授業の改善に役立っているのか？[研究ノート]

評価には総括的評価と形成的評価がある。我が国における学生による授業評価は，当該科目のすべての授業が終了した時点でその授業全体を学生に評価させる総括的評価として実施されることが多い。本研究では，学生による授業評価をより“形成的” にするため，科目が開講されている途中で学生に複数回授業評価を求め，それが授業の改善に役立っているかどうかについて検討した。2 つの講義科目（「教育心理学」と「行動分析学」）の受講生を対象に，授業内容・授業方法を問う10 の質問項目からなる授業評価を複数回実施したところ，平均評価値は授業回数が進むにつれて増加傾向を示した。科目開講中に学生による授業評価を実施することは，「授業内容や授業方法が学生の希望する方向に変化した」という意味において授業の改善に有効であったと考えられる