Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases.Comment: 19 pages, LaTe

Supermembrane in D=5: component action

Based on the connection between partial breaking of global supersymmetry, coset approach, which realized the given pattern of supersymmetry breaking, and the Nambu-Goto actions for the extended objects, we have constructed on-shell component action for N=1, D=5 supermembrane and its dual cousins. We demonstrate that the proper choice of the components and the use of the covariant (with respect to broken supersymmetry) derivatives drastically simplify the action: it can be represented as a sum of four terms each having an explicit geometric meaning.Comment: 14 pages, PACS: 11.30.Pb, 12.60.J

Towards ultrametric theory of turbulence

Relation of ultrametric analysis, wavelet theory and cascade models of turbulence is discussed. We construct the explicit solutions for the nonlinear ultrametric integral equation with quadratic nonlinearity. These solutions are built by means of the recurrent hierarchical procedure which is analogous to the procedure used for the cascade models of turbulence.Comment: 11 page