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We discuss the Nckel model of an open quantum dot, i.e. a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, there are embedded eigenvalues which turn into resonances if the symmetry is violated, either by applying a magnetic field or by deformation of the well. For a weak symmetry breaking we derive the perturbative expansion of these resonances. We also deduce a sufficient condition under which the discrete spectrum of such a system (without any symmetry) survives the presence of a strong magnetic field. It is satisfied, in particular, if the dot potential is purely attractive.

We introduce an irrational factor of order <i>k</i> defined by I k ( n ) = i = 1 l p i i , where I k ( n ) = i = 1 l p i i is the factorization of <i>n</i> and $${\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i &lt; k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}$$. It turns out that the function I k ( n ) = i = 1 l p i i well approximates the characteristic function of <i>k</i>-free integers. We also derive asymptotic formulas for I k ( n ) = i = 1 l p i i and I k ( n ) = i = 1 l p i i .

We propose and analyze a multivariate encryption scheme that uses odd characteristic and an embedding in its construction. This system has a very simple core map F(X) = X^2, allowing for efficient decryption. We also discuss ways to make this decryption faster with specific parameter choices. We give heuristic arguments along with experimental data to show that this scheme resists all known attacks.