Recurrence relations for the number of solutions of a class of Diophantine equations

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all three types of the recursion admit an explicit solution in the form of complete Bell polynomial, depending on the coefficients of the power series expansion of the logarithm of the generating functions for the sequences of individual terms in the Diophantine equations.Comment: 11 pages, Latex. Dedicated to the 70-th anniversary of Apolodor Radut

Probing in-medium vector mesons by dileptons at heavy-ion colliders

An introduction to physics of in-medium hadrons with special emphasis towards modification of vector meson properties in dense nuclear matter is given. We start from remarkable analogy between the in-medium behavior of atoms in gases and hadrons in nuclear matter. Modifications of vector meson widths and masses can be registered experimentally in heavy-ion collisions by detecting dilepton spectra from decays of nucleon resonances and light unflavored mesons including $\rho$- and $\omega$-mesons. Theoretical schemes for description of the in-medium hadrons are reviewed and recent experimental results of the NA60 and HADES collaborations on the dilepton production are discussed.Comment: 18 pages WS LaTeX, 9 eps figures. Lecture given at the 42-nd Predeal International Summer School in Nuclear Physics "Collective Motion and Phase Transitions in Nuclear Systems", 28 August - 9 September, Predeal, Romani