INTRODUCTION Progression of joint destruction in rheumatoid arthritis (RA) is partly heritably; 45 to 58% of the variance in joint destruction is estimated to be explained by genetic factors. The binding of RANKL (Receptor Activator for Nuclear Factor κ B Ligand) to RANK results in the activation of TRAF6 (tumor necrosis factor (TNF) receptor associated factor-6), and osteoclast formation ultim...

‎let $v$ be an $n$-dimensional complex inner product space‎. ‎suppose‎ ‎$h$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎hrightarrow mathbb{c} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎denote by $v_{chi}(h)$ the symmetry class‎ ‎of tensors associated with $h$ and $chi$‎. ‎let $k(t)in‎ (v_{chi}(h))$ be the operator induced by $tin‎ ‎text{end}(v)$‎. ‎the...

In this work, we define the joint normal eigenvalue, joint reducing approximate eigenvalue and we obtain some important and new results about the boundary of joint numerical range of n-tuple operators on a complex Hilbert space.

We consider diagonal cubic surfaces defined by an equation of the form ax + by + cz + dt = 0. Numerically, one can find all rational points of height 6 B for B in the range of up to 10 , thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the author...