In this paper, we study Leibniz algebras with a non-degenerate Leibniz- symmetric fl-invariant bilinear form B, such a pair （g, B） is called a quadratic Leibniz algebra. Our first result generalizes the notion of double extensions to quadratic Leibniz algebras. This notion was introduced by Medina and Revoy to study quadratic Lie alge- bras. In the second theorem, we give a sufficient condition for a quadratic Leibniz algebra to be a quadratic Leibniz algebra by double extension.
This paper provides a fast algorithm for Grobner bases of homogenous ideals of F[x, y] over a finite field F. We show that only the S-polynomials of neighbor pairs of a strictly ordered finite homogenours generating set are needed in the computing of a Grobner base of the homogenous ideal. It reduces dramatically the number of unnecessary S-polynomials that are processed. We also show that the computational complexity of our new algorithm is O(N2), where N is the maximum degree of the input generating polynomials. The new algorithm can be used to solve a problem of blind recognition of convolutional codes. This problem is a new generalization of the important problem of synthesis of a linear recurring sequence.
LI JunBo1, 2 , SU YuCai3 & ZHU LinSheng21 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China 2 Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China 3 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
