Broadband transmission properties of multilayered structures

The formalism of the scattering matrix is applied to describe the transmission properties of multilayered structures with deep variations of the refractive index and arbitrary arrangements of the layers. We show that there is an exact analytical formula for the transmission spectrum, which is valid for the full spectral range and which contains only a limited number of parameters for structures satisfying the quarter-wave condition. These parameters are related to the poles of the scattering matrix, and we present an efficient algorithm to find them, which is based on considering the ray propagation inside the structure and subsequent application of the harmonic inversion technique. These results are significant to analyze the reshaping of ultrashort pulses in multilayered structures.Comment: 3 pages, 3 figure

Public-key cryptography and invariant theory

Public-key cryptosystems are suggested based on invariants of groups. We give also an overview of the known cryptosystems which involve groups.Comment: 10 pages, LaTe

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed. We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is $NP$-hard, and this problem for arbitrary number of variables belongs to $NP$. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series