a watermarking scheme for grayscale image isproposed based on a family of the chaotic maps and discretecosine transform. jacobian elliptic mapis employed to encrypt ofwatermarked logo. piecewise nonlinear chaotic map is also usedto determine the location of dct coefficients for the watermarkembedding. the purpose of this algorithm is to improve theshortcoming of watermarking such as small key s...

Any endomorphism of Cn defined by n polynomials with everywhere non-vanishing Jacobian is an automorphism. The Jacobian conjecture originated fromKeller ([5]). Let F1, . . . , Fn ∈ C[x1, . . . , xn] be a set of n polynomials in n variables with n ≥ 1 such that the Jacobian of these polynomials is a nonzero constant. The Jacobian conjecture says that the subalgebra C[F1, . . . , Fn] of C[x1, . ....

J ‘ edrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomial...

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the Jacobian set – the set of all algebra endomorphisms of a polynomial algebra with the Jacobian 1 – the Jacobian conjecture claims ...

The transition from quasiperiodicity to chaos is studied in a two-dimensional dissipative map with the inverse golden mean rotation number. On the basis of a decimation scheme, it is argued that the (minimal) slope of the critical iterated circle map is proportional to the effective Jacobian determinant. Approaching the zero-Jacobian-determinant limit, the factor of proportion becomes a univers...

Suppose f(x, y), g(x, y) are two polynomials with complex coefficients. The classical Jacobian Conjecture (due to Keller) asserts the following. Conjecture. (Jacobian Conjecture in dimension two) If the Jacobian of the pair (f, g) is a non-zero constant, then the map (x, y) 7→ (f(x, y), g(x, y)) is invertible. Note that the opposite is clearly true, because the Jacobian of any polynomial map is...

The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for C maps is explored here. Some results known in the polynomial case are extended to the C context, and some special cases are resolved.

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the corresponding Abel map AL:C → J̄ , which maps C into its compactified Jacobian, and form its pullback map A L : Pic J̄ → J , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then A L is known to be an isomorphis...