Perturbations of moving membranes in AdS_7

We study the stability of uniformly moving membrane-like objects in seven dimensional Anti-de Sitter space. This is approached by a linear perturbation analysis and a search for growing modes. We examine both analytic and numerical configurations previously found in [1].Comment: 20 pages, 6 figure

Continued Fractions by the Nearest Even Number. The short type

Abstract: In the preprint «Continued fractions by the nearest even number», we propose a new type of one-dimensional continued fractions. It has many properties of the classic continued fractions: convergence, uniqueness of expansion, periodicity for quadratic irrationalities. But they converge very slow. Here we propose a fast (short) type of even continued fractions. The modi»ed algorithm allows to replace a sequence of similar parts of even continued fraction by one linear fractional transformation.Note: Research direction:Mathematical problems and theory of numerical method

Klein's Polyhedra for the Seventh Extremal Cubic Form.

Abstract: Davenport and Swinnerton-Dyer had found the first 19 extremal thernar cubic forms githe meaning of which is the same as the meaning of the Markov forms for binary quadratic case. The Klein's polyhedra for the forms g1-g6were recently computed by Bruno and Parusnikov. For the multiple vectors of these forms, they have computed convergents of continued fractions for some matrix generalizations of the continued fractions algorithm as well. In this paper the analogious problems for the form g7are studied. Namely, the Klein polyhedra for the form g7are computed. Their periods and fundamental domains are found. The matrix algorithm's expansions of the multiple vector of this form are computed as well.Note: Research direction:Mathematical problems and theory of numerical method

Klein's Polyhedra for the Fifth Extremal Cubic Form.

Abstract: Davenport and Swinnerton-Dyer had found the first 19 extremal thernar cubic forms gi, the meaning of which is the same as the meaning of the Markov forms for binary quadratic case. The Klein's polyhedra for the forms g1-g5were recently computed by Bruno and Parusnikov. For the multiple vectors of these forms, they have computed convergents of continued fractions for some matrix generalizations of the continued fractions algorithm as well. In this paper the analogious problems for the form g6are studied. Namely, the Klein polyhedra for the form g6are computed. Their periods and fundamental domains are found. The matrix algorithm's expansions of the multiple vector of this form are computed as well.Note: Research direction:Mathematical problems and theory of numerical method

Polyhedra of Absolute Values for Triples of Linear Forms.

Abstract: In the preprint 'The correct generalization of the continued fraction' by A.D.Bruno (No. 86/2003) there were proposed to a set of three homogenous linear forms to put in correspondence two polyhedral surfaces in the space of absolute values of these forms. The generalization of the continued fraction algorithm is a motion along vertices of one of these surfaces. Here in pictures we give results of computation of the surfaces for eleven cubic forms being products of three linear forms. They show periodic structures and confirm the correctness of the generalized algorithm.Note: Research direction:Mathematical problems and theory of numerical method