Solutions of Indefinite Equations

Indefinite equation is an unsolved problem in number theory. Through explo-ration, the author has been able to use a simple elementary algebraic method to solve the solutions of all three variable indefinite equations. In this paper, we will introduce and prove the solutions of Pythagorean equation, Fermat’s the-orem, Bill equation and so on.

SYMBOLIC VERSOR COMPRESSION ALGORITHM

In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.

ACCURATE ATTENUATION CORRECTION FOR ALGEBRAIC RECONSTRUCTION TECHNIQUE IN SPECT

We present a new iterative reconstruction algorithm to improve the algebraic reconstruction technique （ART） for the Single-Photon Emission Computed Tomography. Our method is a generalization of the Kaczmarz iterative algorithm for solving linear systems of equations and introduces exact and implicit attenuation correction derived from the attenuated Radon transform operator at each step of the algorithm. The performances of the presented algorithm have been tested upon various numerical experiments in presence of both strongly non-uniform attenuation and incomplete measurements data. We also tested the ability of our algorithm to handle moderate noisy data. Simulation studies demonstrate that the proposed method has a significant improvement in the quality of reconstructed images over ART. Moreover, convergence speed was improved and stability was established, facing noisy data, once we incorporate filtration procedure in our algorithm.

Three New(2+1)-dimensional Integrable Systems and Some Related Darboux Transformations

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1,then under the framework of zero curvature equations we generate two(2+1)-dimensional integrable hierarchies, including the(2+1)-dimensional shallow water wave(SWW) hierarchy and the(2+1)-dimensional Kaup–Newell(KN)hierarchy. Through reduction of the(2+1)-dimensional hierarchies, we get a(2+1)-dimensional SWW equation and a(2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the(2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the(2+1)-dimensional KN equation could be deduced. Finally,with the help of the spatial spectral matrix of SWW hierarchy, we generate a(2+1) heat equation and a(2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang–Mills equations.