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The present paper focuses on the theory of two-dimensional (2D) linear cellular automata (CA) withrespect to uniform reflexive and periodic boundary conditions. It is investigated the theoretical aspects of 2D linearCA over binary states field with image problem. We consider geometrical and visual aspects of images generatedby these CA transition rules. Multiple copies of any arbitrary images corresponding to CA can be studied furtherby considering theses transition rules of von Neumann and Moore CAs. An important note that these special typesof CAs can be applied many different special problems e.g. computability theory, applied mathematics, theoreticalchemistry and biology, DNA and genetics research, image science, textile design.

In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra and the ideal generated by the squares of elements (further denoted by I) is a right -module. Using description (Casati et al J. Math. Phys. 51: 033515 (2010)) of representations of algebra in and where or we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra Classification of Leibniz algebras with corresponding Lie algebra and with the ideal I as a Fock right -module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces B L (2) and Z L (2) are constructed, as well.

We study the phase diagram for the Ising model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J(1), prolonged next-nearest-neighbor interactions J(p), and one-level k-tuple neighbor interaction J(o). The phase diagram is studied for several ranges of the competing parameters; it shows the appearance of several features and modulated phases arising from the frustration effects introduced by the one-level k-tuple neighbor interaction J(o). The variation of the wayevector with temperature in the modulate phase is studied in detail; it shows narrow commensurate steps between incommensurate regions. Finally, the Lyapunov exponent associated with the trajectory of the system is investigated. (C) 2011 Elsevier B.V. All rights reserved.

We consider a generalization of eight-dimensional multiply warped product manifolds as a special warped product, by allowing the fiber metric to be non-block diagonal. We define this special warped product as a (3 + 3 + 2) warped-like manifold of the form M = F x B. where the base B is a two-dimensional Riemannian manifold, and the fibre F is of the form F = F-1 x F-2 where the F-i(i = 1, 2) are Riemannian 3-manifolds. We prove that the connection on M is completely determined by the requirement that the Bonan 4-form given in the work of Yasui and Ootsuka [Y. Yasui and T. Ootsuka, Spin(7) holonomy manifold and superconnection, Class. Quantum Gravity 18(2001)807-816] be closed. Assuming that the F-i are complete, connected and simply connected, it follows that they are isometric to S-3 with constant curvature k > 0 and the Yasui-Ootsuka solution is unique in the class of (3 + 3 + 2) warped-like product metrics admitting a specific Spin(7) structure. Crown Copyright (C) 2011 Published by Elsevier B.V. All rights reserved.

The Potts model on the Cayley tree of second order, with competing interactions between the nearest-neighbor, pro longed next-nearest neighbor and two-level triple neighbors has been studied. The phase diagram is studied for several ranges of the competing parameters showing the appearance of several features and modulated phases arising from the frustration effects introduced by the two-level triple neighbors interaction.

We study the phase diagram for Ising model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J(1) and prolonged next-nearest-neighbor interactions J(p). To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Vannimenus for k = 2.