Badly approximable vectors on a vertical Cantor set

For $i, j > 0, i + j = 1$, the set of badly approximable vectors with weight $(i, j)$ is defined by $Bad(i, j) = \{(x, y) \in \R^2 : \exists c > 0 \forall q\in\N, \;\; \max\{q||qx||^{1/i}, q||qy||^{1/j} \} > c\}$, where $||x||$ is the distance of $x$ to the nearest integer. In 2010 Badziahin-Pollington-Velani solved Schmidt's conjecture which was stated in 1982, proving that $Bad(i, j) \cap Bad(j, i)$ is nonempty. Using Badziahin-Pollington-Velani's technique with reference to fractal sets, we were able to improve their results: Assume that we are given a sequence $(i_t, j_t)$ with $i_t, j_t > 0, i_t + j_t = 1$. Then, the intersection of $Bad(i_t, j_t)$ over all t is nonempty

B Physics and Quarkonia studies with early ATLAS data

Quarkonia and B-Physics are among the first areas to be investigated with the first data collected by ATLAS. The ATLAS detector at CERN's LHC is preparing to take data from proton-proton collisions expected to start by the end of 2009. Investigation of the decay of B-hadrons represents a complementary approach to direct searches for Physics beyond the Standard Model. Early B-physics data will provide valuable information on the detector performance, as well as allow calibration studies in support of new Physics searches. Meaningful quarkonia studies performed with early data are expected to have the reach to make authoritative statements about the underlying production mechanism and provide cross-sections in this new energy regime. We review various aspects of prompt quarkonium production at the LHC: the accessible ranges in transverse momentum and pseudo-rapidity, spin alignment of vector states, separation of color octet and color singlet production mechanism and feasibility of observing radiative chi_c decaysComment: Proceedings of the Fourth High-Energy Physics International Conference HEP-MAD 09, Antananarivo, Madagascar, August 21-28, 200

IVF Battles: Legal Categories and Comparative Tales

Coupled with modern reproductive technologies, the ancient desire for parenthood has led to novel legal challenges. This essay discusses landmark cases addressing those challenges. At the outset, it distinguishes between two litigation paradigms in this area—termed “horizontal” and “vertical.” Horizontal controversies involve private parties who have different aspirations regarding a joint parenthood project (e.g., between two partners who began an IVF procedure and later disagree whether to complete the process). In contrast, vertical controversies concern clashes between an individual (or individuals) and the state, such as when the state or one of its authorities does not allow the individual to move forward with technologies that may lead to parenthood (e.g., new surrogacy procedures), though all affected individuals consent. The essay then focuses on horizontal litigation, and examines the ways in which various legal systems draw on, and sometimes adjust to the particular circumstances of the case, traditional concepts such as contract, reliance, property, and more to resolve such disputes