Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings

We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary solution) then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, generalizes the classical operator factorization in the linear context. Sequences of ratios of consecutive terms of a unitary solution are used to obtain the semiconjugate factorization. Such sequences, known as eigensequences are well-suited to variable coefficients; for instance, they provide a natural context for the expression of the classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear difference equations with periodic coefficients and also derive formulas for the general solutions of linear functional recurrences satisfied by the classical special functions such as the modified Bessel and Chebyshev.Comment: Application of nonlinear semiconjugate factorization theory to linear difference equations with variable coefficients in rings; 29 pages, containing the main theory and more than 8 examples worked out in detai

CERT strategy to deal with phishing attacks

Every day, internet thieves employ new ways to obtain personal identity people and get access to their personal information. Phishing is a somehow complex method that has recently been considered by internet thieves.The present study aims to explain phishing, and why an organization should deal with it and its challenges of providing. In addition, different kinds of this attack and classification of security approaches for organizational and lay users are addressed in this article. Finally, the CERT strategy is presented to deal with phishing and studying some anti-phishing