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

Emphysematous Cystitis in a Patient with Type 2 Diabetes Mellitus

A 62-year-old woman with a history of poorly controlled type 2 diabetes mellitus was admitted to our hospital with a 3-week history of mild fever, vomiting, and anorexia. Abdominal computed tomography (CT) showed bilateral hydronephrosis and gas accumulation in the urinary bladder wall and left ureter. Laboratory tests showed leukocytosis and elevated C-reactive protein level. Urine culture showed heavy growth of Escherichia coli. The final diagnosis was emphysematous cystitis. The patient was treated with systemic antibiotics and drainage using a urethral catheter. The clinical and radiographic findings resolved rapidly, and she was discharged from the hospital on day 28. Emphysematous cystitis is a relatively rare urinary tract infection associated with gas formation, and has the potential for a serious outcome if untreated. Early detection by imaging studies such as CT is important in providing prompt treatment and favorable clinical outcome

ON THE NAGELL-LJUNGGREN EQUATION (Analytic Number Theory : Distribution and Approximation of Arithmetic Objects)

We show that there exists an effective upper bound for the solutions to the Nagell-Ljunggren equation of the form --=y^{q} in 4 unknowns in integers x>1, y>1, m>2, q>1, when x is a cube of an integer. Our method relies on a refined estimate of linear forms in logarithms

Characterization of Mouse Tissue Kallikrein 5

Mouse tissue kallikreins (Klks) are members of a large, multigene family consisting of 37 genes, 26 of which can code for functional proteins. Mouse tissue kallikrein 5 (KIk5) has long been thought to be one of these functional genes, but the gene product, mK5, has not been isolated and characterized. In the present study, we prepared active recombinant mK5 using an Escherichia coli expression system, followed by column chromatography. We then determined the biochemical and enzymatic properties of purified mK5. mK5 had trypsin-like activity for Arg at the P1 position, and its activity was inhibited by typical serine protease inhibitors. mK5 degraded gelatin, fibronectin, collagen type IV, high-molecular-weight kininogen, and insulin-like growth factor binding protein-3. Our data suggest that mK5 may be implicated in the process of extracellular matrix remodeling

Monitoring mature tomato (red stage) quality during storage using ultraviolet-induced visible fluorescence image

The potential of UV-induced fluorescence imaging was investigated as a non-destructive tool to monitor postharvest quality degradation of tomatoes harvested at the red stage and stored at 25 °C. The fluorescence images (excitation at 365 nm) were found to be a better indicator of tomato quality degradation than color images after color saturation. Tomatoes were stored at 25 °C for 9 d. The changes in color and fluorescence of tomato were evaluated by two types of images: Color and fluorescence images. A conventional colorimeter was also used for as a reference. Changes in the RGB ratio for these two types of images were opposite. In the color images, the G ratio decreased rapidly for the initial 3 or 5 d and then stabilized afterwards. On the other hand, in the fluorescence images, the G ratio increased continuously up to 9 d. Given that temperature conditions during transportation and storage of tomatoes is not always ideal, the results from this research provide the foundation for developing a postharvest monitoring system of mature tomato quality degradation