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

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

Two regulatory steps of ER-stress sensor Ire1 involving its cluster formation and interaction with unfolded proteins

Chaperone protein BiP binds to Ire1 and dissociates in response to endoplasmic reticulum (ER) stress. However, it remains unclear how the signal transducer Ire1 senses ER stress and is subsequently activated. The crystal structure of the core stress-sensing region (CSSR) of yeast Ire1 luminal domain led to the controversial suggestion that the molecule can bind to unfolded proteins. We demonstrate that, upon ER stress, Ire1 clusters and actually interacts with unfolded proteins. Ire1 mutations that affect these phenomena reveal that Ire1 is activated via two steps, both of which are ER stress regulated, albeit in different ways. In the first step, BiP dissociation from Ire1 leads to its cluster formation. In the second step, direct interaction of unfolded proteins with the CSSR orients the cytosolic effector domains of clustered Ire1 molecules

Isoscalar Giant Quadrupole Resonance State in the Relativistic Approach with the Momentum-Dependent Self-Energies

We study the excited energy of the isoscalar giant quadrupole resonance with the scaling method in the relativistic many-body framework. In this calculation we introduce the momentum-dependent parts of the Dirac self-energies arising from the one-pion exchange on the assumption of the pseudo-vector coupling with nucleon field. It is shown that this momentum-dependence enhances the Landau mass significantly and thus suppresses the quadrupole resonance energy even giving the small Dirac effective mass which causes a problem in the momentum-independent mean-field theory.Comment: 12pages, 2 Postscript figure

Non-drug and surgical treatment algorithm and recommendations for the 2020 update of the Japan College of Rheumatology clinical practice guidelines for the management of rheumatoid arthritis—secondary publication

[Objectives] The aim of this study was to update the Japan College of Rheumatology (JCR) clinical practice guidelines (CPGs) for the management of rheumatoid arthritis (RA) and prepare an algorithm for non-drug and surgical treatments. This article is a digest version of the guidelines. [Methods] The Japanese Ministry of Health, Labour and Welfare’s research group, in collaboration with the JCR, used the Grading of Recommendations, Assessment, Development, and Evaluation method to update the 2014 JCR CPG for RA. The consensus was formed by CPG panel members. [Results] We raised 19 clinical questions regarding non-drug and surgical treatments for RA and developed recommendations. The treatments included exercise therapy; occupational therapy; joint injection of corticosteroids; and orthopaedic surgeries including cervical spine surgery, wrist and foot arthroplasty, ankle arthrodesis, and replacement arthroplasty of the shoulder, elbow, finger, hip, knee, and ankle. Recommendations regarding the risks of surgery and perioperative discontinuation of medications have also been developed. Based on these recommendations, we created an original algorithm for the non-drug and surgical treatment of RA. [Conclusions] These recommendations are expected to serve rheumatologists, health care professionals, and patients with RA as tools for shared decision-making to treat residual limb joint symptoms and functional impairment

\'Etude du cas rationnel de la th\'eorie des formes lin\'eaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u\in Lie(G(C_{v_{0}})) a logarithm of a point p\in G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)\otimes_{k} C_{v_{0}}. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height Log a of p, removing a polynomial term in LogLog a. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of J.-B. Bost) obtained from a lemma by M. Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by D. Roy.Comment: Version d\'efinitiv