5,242 research outputs found
Tropical Effective Primary and Dual Nullstellens\"atze
Tropical algebra is an emerging field with a number of applications in
various areas of mathematics. In many of these applications appeal to tropical
polynomials allows to study properties of mathematical objects such as
algebraic varieties and algebraic curves from the computational point of view.
This makes it important to study both mathematical and computational aspects of
tropical polynomials.
In this paper we prove a tropical Nullstellensatz and moreover we show an
effective formulation of this theorem. Nullstellensatz is a natural step in
building algebraic theory of tropical polynomials and its effective version is
relevant for computational aspects of this field.
On our way we establish a simple formulation of min-plus and tropical linear
dualities. We also observe a close connection between tropical and min-plus
polynomial systems
Continuous dielectric permittivity I: Specific features of the dielectric continuum solvation model with a position-dependent permittivity function
We consider a modified formulation for the recently developed new approach in
the continuum solvation theory (Basilevsky, M. V., Grigoriev, F. V., Nikitina,
E. A., Leszczynski, J., J. Phys. Chem. B 2010, 114, 2457), which is based on
the exact solution of the electrostatic Poisson equation with the
space-dependent dielectric permittivity. Its present modification ensures the
property curl E = 0 for the electric strength field E inherent to this
solution, which is the obligatory condition imposed by Maxwell equations. The
illustrative computation is made for the model system of the point dipole
immersed in a spherical cavity of excluded volume.Comment: 31 pages, 4 figure
Yao’s millionaires’ problem and public-key encryption without computational assumptions
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