The algebraic structure behind the derivative nonlinear Schrödinger equation

The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schrödinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of an Ŝℓ2Kac-Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows. The equivalence between the latter and the massive Thirring model is also explicitly demonstrated. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation. © 2013 IOP Publishing Ltd.

A decoding method of an n length binary BCH code through (n + 1)n length binary cyclic code

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Construction and decoding of BCH codes over chain of commutative rings

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A0 ⊂ A1 ⊂···⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0 ⊂ K1 ⊂···⊂ Kt−1 ⊂ Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A∗ i and K∗ i are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A∗ i and K∗ i for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

Goppa codes through polynomials of B[X;(1/2^2)Z_0] and its decoding principle

In this paper, we present a decoding principle for Goppa codes constructed by generalized polynomials, which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight $t\leq 2r$, i.e., whose minimum Hamming distance is $2^{2}r+1$.

A decoding procedure which improves code rate and error corrections

Corresponding to $C_{0}[n,n-r]$, a binary cyclic code generated by a primitive irreducible polynomial $p(X)\in \mathbb{F}_{2}[X]$ of degree $r=2b$, where $b\in \mathbb{Z}^{+}$, we can constitute a binary cyclic code $C[(n+1)^{3^{k}}-1,(n+1)^{3^{k}}-1-3^{k}r]$, which is generated by primitive irreducible generalized polynomial $p(X^{\frac{1}{3^{k}}})\in \mathbb{F}_{2}[X;\frac{1}{3^{k}}\mathbb{Z}_{0}]$ with degree $3^{k}r$, where $k\in \mathbb{Z}^{+}$. This new code $C$ improves the code rate and has error corrections capability higher than $C_{0}$. The purpose of this study is to establish a decoding procedure for $C_{0}$ by using $C$ in such a way that one can obtain an improved code rate and error-correcting capabilities for $C_{0}$.

Algebraic constructions of densest lattices

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Algebraic solution of an anisotropic nonquadratic potential

We show that an anisotropic nonquadratic potential, for which a path integral treatment has been recently discussed in the literature, possesses the SO(2, 1) ⊗SO(2, 1) ⊗SO(2, 1) dynamical symmetry, and construct its Green function algebraically. A particular case which generates new eigenvalues and eigenfunctions is also discussed. © 1990.

Algebraic solution of an anisotropic ring-shaped oscillator

Using an algebraic technique related to the SO (2, 1) group we construct the Green function for the potential ar2 + b(r sin θ)-2 + c(r cos θ)-2 + dr2 sin2θ + er2 cos2θ. The energy spectrum and the normalized wave functions are also obtained. © 1990.

Review of The Other Alberta: Decoding a Political Enigma by Doreen Barrie

Doreen Barrie should have subtitled this book "Advocating a Different Identity" because this is its basic thrust. In Barrie's view, today's wealthy, modern, and expansive Alberta should abandon its historic grievances and hostility towards Ottawa. Instead, it should embrace a new narrative emphasizing "the positive qualities Albertans possess . . . the contributions the province has made to the country . . . and that Albertans share fundamental Canadian values with people in other parts of Canada and are eager to playa larger role on the national stage."

Does Decoding Increase Word Problem Solving Skills?

In this action research study of my classroom of 7th grade mathematics, I investigated whether the use of decoding would increase the students’ ability to problem solve. I discovered that knowing how to decode a word problem is only one facet of being a successful problem solver. I also discovered that confidence, effective instruction, and practice have an impact on improving problem solving skills. Because of this research, I plan to alter my problem solving guide that will enable it to be used by any classroom teacher. I also plan to keep adding to my math problem solving clue words and share with others. My hope is that I will be able to explain my project to math teachers in my district to make them aware of the importance of knowing the steps to solve a word problem.

Algebraic-graph method for identification of islanding in power system grids

In this paper, a new algebraic-graph method for identification of islanding in power system grids is proposed. The proposed method identifies all the possible cases of islanding, due to the loss of a equipment, by means of a factorization of the bus-branch incidence matrix. The main features of this new method include: (i) simple implementation, (ii) high speed, (iii) real-time adaptability, (iv) identification of all islanding cases and (v) identification of the buses that compose each island in case of island formation. The method was successfully tested on large-scale systems such as the reduced south Brazilian system (45 buses/72 branches) and the south-southeast Brazilian system (810 buses/1340 branches). (C) 2011 Elsevier Ltd. All rights reserved.