15,077 research outputs found
Multipurpose S-shaped solvable profiles of the refractive index: application to modeling of antireflection layers and quasi-crystals
A class of four-parameter solvable profiles of the electromagnetic admittance
has recently been discovered by applying the newly developed Property & Field
Darboux Transformation method (PROFIDT). These profiles are highly flexible. In
addition, the related electromagnetic-field solutions are exact, in closed-form
and involve only elementary functions. In this paper, we focus on those who are
S-shaped and we provide all the tools needed for easy implementation. These
analytical bricks can be used for high-level modeling of lightwave propagation
in photonic devices presenting a piecewise-sigmoidal refractive-index profile
such as, for example, antireflection layers, rugate filters, chirped filters
and photonic crystals. For small amplitude of the index modulation, these
elementary profiles are very close to a cosine profile. They can therefore be
considered as valuable surrogates for computing the scattering properties of
components like Bragg filters and reflectors as well. In this paper we present
an application for antireflection layers and another for 1D quasicrystals (QC).
The proposed S-shaped profiles can be easily manipulated for exploring the
optical properties of smooth QC, a class of photonic devices that adds to the
classical binary-level QC.Comment: 14 pages, 18 fi
How to determine linear complexity and -error linear complexity in some classes of linear recurring sequences
Several fast algorithms for the determination of the linear complexity of -periodic sequences over a finite
field \F_q, i.e. sequences with characteristic polynomial , have been proposed in the literature.
In this contribution fast algorithms for determining the linear complexity of binary sequences with characteristic
polynomial for an arbitrary positive integer , and are presented.
The result is then utilized to establish a fast algorithm for determining the -error linear complexity of
binary sequences with characteristic polynomial
Cyclic Resultants
We characterize polynomials having the same set of nonzero cyclic resultants.
Generically, for a polynomial of degree , there are exactly
distinct degree polynomials with the same set of cyclic resultants as .
However, in the generic monic case, degree polynomials are uniquely
determined by their cyclic resultants. Moreover, two reciprocal
(``palindromic'') polynomials giving rise to the same set of nonzero cyclic
resultants are equal. In the process, we also prove a unique factorization
result in semigroup algebras involving products of binomials. Finally, we
discuss how our results yield algorithms for explicit reconstruction of
polynomials from their cyclic resultants.Comment: 16 pages, Journal of Symbolic Computation, print version with errata
incorporate
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