Abstract: The circular watermarking (CW) technique has attracted increasing attention because it can resist the estimation of secret carriers in the watermarked only attack (WOA) framework. However, the existing CW schemes are not applicable whenever a malicious watermark removal attack can take place. This is because they either have low security because the attacker can disclose the embedding subspace or have low robustness. Based on an existing CW scheme called transportation natural watermarking (TNW), this correspondence presents a new CW technique for the tradeoff between robustness and security, which we refer to as controllable secure watermarking (CSW). The idea behind the CSW is that by altering the host signal in the orthogonal complement of the embedding subspace, we can make the watermarked signal have an orthogonally invariant distribution in a higher dimensional subspace including the embedding subspace. Orthogonally invariant distribution essentially requires that the distribution does not change if multiplied by any freely chosen orthogonal matrix, and the higher dimensional subspace is referred to as invariant subspace. We prove that the attacker can only reduce the uncertainty of secret carriers up to the invariant subspace. The dimension of the invariant subspace can be used for the tradeoff between robustness and security. Further, the experiment results show that the robustness-security tradeoff provided by the CSW is efficient. In particular, with the increase of the dimension of the invariant subspace, the security of the CSW will increase quickly while its robustness will only decrease slowly.

Abstract: We study the geometry of lightlike hypersurfaces M of an inde nite cosymplectic manifold such that either (1) the characterist vector field of belongs to the screen distribution S(TM) of M or (2) belongs to the orthogonal complement of S(TM) in

Abstract: We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.

Abstract: The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of governing differential equations for variational problems on the space of conformal vector fields. Several examples are given. In addition, the paper also gives a review, in full detail, of already known vector field decompositions involving subalgebras of volume preserving and symplectic vector fields.

Abstract: Diagnostic modeling based on computational anatomy is an important topic. In previous work, discrimination method using support vector machine based on principal component analysis of the hippocampus shapes have been proposed. However, disease-specific component was not considered explicitly. In this paper, we propose a method for constructing the disease subspace using orthogonal complement of the normal subspace. The proposed method was tested using the hepatic cirrhosis and hip osteoarthritis datasets and was compared to a previous method. In our experiments, the proposed method was effective for disease discrimination based on organ shapes.

Abstract: We present sampling theorems that reconstruct consistent signals from noisy underdetermined measurements. The consistency criterion requires that the reconstructed signal yields the same measurements as the original one. The main issue in underdetermined cases is a choice of a complementary subspace L in the reconstruction space of the intersection between the reconstruction space and the orthogonal complement of the sampling space because signals are reconstructed in L. Conventional theorems determine L without taking noise in measurements into account. Hence, the present paper proposes to choose L such that variance of reconstructed signals due to noise is minimized. We first arbitrarily fix L and compute the minimum variance under the condition that the average of the reconstructed signals agrees with the noiseless reconstruction. The derived expression clearly shows that the minimum variance depends on L and leads us to a condition for L to further minimize the minimum value of the variance. This condition indicates that we can choose such an L if and only if L includes a subspace determined by the noise covariance matrix. Computer simulations show that the standard deviation for the proposed sampling theorem is improved by 8.72% over that for the conventional theorem.

Abstract: Consider the eigenvalue problem QAQ~x = λ~x, where A is an n×n matrix and Q is projection matrix onto a subspace S⊥ of dimension n−k. In this paper we construct a meromorphic function whose zeros correspond to the (generically) n−k nonzero eigenvalues of QAQ. The construction of this function requires only that we know A and a basis for S, the orthogonal complement of S⊥. The formulation of the function is assisted through the use of the Frobenius inner product; furthermore, this inner product allows us to directly compute the eigenvalue when k = 1 and n = 2. When n = 3 and k = 1 we carefully study four canonical cases, as well as one more general case.

Abstract: We consider finite energy and $L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion harmonicity for finite energy 1-forms. As as corollary, under a certain capacity-separation assumption, we prove that the space of harmonic 1-forms is nontrivial if and only if the classical Cech cohomology is nontrivial. In the examples of classical self-similar fractals these spaces typically are either trivial or infinitely dimensional. Finally, we study Navier-Stokes type models and prove that under our assumptions they have only steady state divergence-free solutions. In particular, we solve the existence and uniqueness problem for the Navier-Stokes and Euler equations for a large class of fractals that are topologically one-dimensional but can have arbitrary Hausdorff and spectral dimensions.