A Mathematical Framework for Active Steganalysis by Chandramouli By Chandramouli

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Eliasson ε1 ∼ F1 + D1 ∈ N F(α1 , . . , ρ1 ; Ω1 , P1 ) ⇓ ⇓ ⇐⇐⇐ T (σ1 , s) & ⇓ ⇓ C(λ2 , µ2 , ν2 ) − clustering ⇓ ⇓ ⇓ ε2 ∼ F2 + D2 ∈ N F(α2 , . . , ρ2 ; Ω2 , P2 ) ⇒⇒⇒ & ⇓ ⇓ ⇐⇐⇐ T (σ2 , s) ⇓ ⇓ C(λ3 , µ3 , ν3 ) − clustering ⇓ ⇓ ⇓ ε3 ∼ F3 + D3 ∈ N F(α3 , . . 1) where V is a real valued function on the one-dimensional torus T = R/(2πZ) and ω is a real number. We assume that V is piecewise Gevrey smooth on a partition with p many pieces and | V |C k ≤ βγ k ∀k ≥ 0. 2) The functions V (θ + y) − V (θ)) satisﬁes for all θ, y the two transversality conditions max0≤k≤s | ∂yk (V (θ + y) − V (θ)) | ≥ σ max0≤k≤s | ∂θk (V (θ + y) − V (θ)) | ≥ σ y .

H. Eliasson Then the vectors I 0 R and I will span Λ1 and Λ2 respectively. For a matrix M denote by Mi the matrix whose (k, l)-entry is Mkl if l−k = i and 0 otherwise. Then S = S0 + · · · + Sm1 , T = T0 + · · · + Tm2 , A = A−m1 + · · · + Am2 . The equation for R, SR − RT = −A, can now be written S0 Rj − Rj T0 = −Aj − Sk Rj−k + k≥1 Rj−l Tl l≥1 for −m1 ≤ j ≤ m2 . From this we get that β | R | ≤ constm( )m+1 . r It follows from this by Gram-Schmidt that we get an ON-basis for Λ2 A B where B is a triangular m2 × m2 -matrix whose diagonal entries have absolute value 1 r ≥ const ( )m+1 , m β and | B | satisﬁes the same estimate as | R |.

Notice that if λ = 0 is an eigenvalue of AB, then it is also an eigenvalue of BA, with the same multiplicity. Hence, the spectrum of AA∗ and A∗ A is the same except possibly for an eigenvalue 0. Now the kernel of A∗ A is precisely the kernel of A, so if A is an k × l-matrix of rank r then the dimension of the kernel is l − r. Since the rank of A and A∗ is the same, we get that 0 is an eigenvalue of AA∗ and A∗ A of multiplicity k − r and l − r respectively. I A and Λ2 = , then If now Λ1 = 0 B Λ∗1 Λ2 Λ∗2 Λ1 = AA∗ , Λ∗2 Λ1 Λ∗1 Λ2 = A∗ A, so the ﬁrst ﬁrst statement is proved.