New (virtual) Physics in the Era of the LHC

A simple extension of the Standard Model demonstrates that New Physics non-reachable through direct production at LHC can induce up to 10% corrections to the length of the unitarity triangle side, extracted from $\Delta m_{B_d}$.Comment: 5 page

Photon "mass" and atomic levels in a superstrong magnetic field

The structure of atomic levels originating from the lowest Landau level in a superstrong magnetic field is analyzed. The influence of the screening of the Coulomb potential on the values of critical nuclear charge is studied.Comment: Contribution to the proceedings of the LOWDIM Workshop devoted to Sergei Gaikovich Matinyan 80'th birthday; 6 pages, 2 figure

The rise and fall of the fourth quark-lepton generation

The existence of the fourth quark-lepton generation is not excluded by the electroweak precision data. However, the recent results on the 126 GeV higgs boson production and decay do not allow an extra generation at least as far as the perturbation theory can be used.Comment: Report at the conference QTHEP 2013, Repino, June 23-30 2013 and lecture at the 2013 Helmholz School Physics of Heavy Quarks and Hadrons, Dubna, July 15-28. An important acknowledgement is adde

On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve passing through these points is at least half of the perimeter of $K$. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of $K$. Moreover, the factor $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to the perimeter of a convex shape.Comment: 7 pages, 8 figure