nearly zero. (3) A watermark is randomized so that a pattern of cyclic cross stripes may disappear and an embedded watermark may be inconspicuous. The embedded watermark is extracted by multiplying the conversely randomized image embedding a watermark by the random sequence. (4) An ID pattern is extracted by summing up the products of any part of the random sequence and the corresponding part o...

Consider a N × n random matrix Yn = (Y n ij ) where the entries are given by Y n ij = σ(i/N,j/n) √ n Xn ij , the X n ij being centered i.i.d. and σ : [0, 1] 2 → [0,∞) being a continuous function called a variance profile. Consider now a deterministic N ×n matrix Λn = ( Λij ) whose off-diagonal entries are zero. Denote by Σn the non-centered matrix Yn+Λn and by N ∧n = min(N,n). Then under the as...

We study the existence of fully nontrivial solutions to system ??ui+?iui=?j=1??ij|uj|p|ui|p?2uiin?,i=1,…,?,in a bounded or unbounded domain ? in RN, N?3. The ?i’s are real numbers, and nonlinear term may have subcritical (1NN?2). matrix (?ij) is symmetric admits block decomposition such that diagonal entries ?ii positive, interaction forces w...

We show that solutions of the Bach equation exist which are not conformal Einstein spaces. In connection with fourth order gravitational field equations, cf. e.g. [1, 2] where the breaking of conformal invariance was discussed, the original Bach equation, Bij = 0 , (1) enjoys current interest. Eq. (1) stems from a Lagrangian L = 1 2 √ −gCijklC , and variation gives, cf. Bach [3], 1 √−g δL/δg ij...

We discuss iterative methods for the solution of the linear system Ax = b, which are based on a single splitting or a multisplitting of A. In order to compare diierent methods , it is common to compare the spectral radius of the iterative matrix. For M{matrices A and weak regular splittings there exist well-known comparison theorems. Here, we give a comparison theorem for splittings of Hermitia...

In this paper, we introduce and study the notion of regular generalized –γ– closed sets. Also, some of their properties are investigated. Further, the notion of regular generalized –γ– open sets and the notion of regular generalized –γ– neighbourhood are discussed. Moreover, some of its basic properties are introduced. Finally, we give an applications of rgγ–closed sets, say γ–regular 1 2 T spa...

Let G = a 1 ,. .. , a n | a i a j a i · · · = a j a i a j. .. , i < j be an Artin group and let m ij = m ji be the length of each of the sides of the defining relation involving a i and a j. We show if all m ij ≥ 7 then G is relatively hyperbolic in the sense of Farb with respect to the collection of its two-generator subgroups a i , a j for which m ij < ∞.