Three-dimensional point-cloud data has been enormously abundant with the emergence of 3D data acquisition, processing, and visualization technologies. Encryption algorithms have been recently ...introduced to ensure secure storage and communication for this type of data. Maintaining algorithmic correctness and the geometric stability are still key challenges towards the construction of reliable, trustful, and practical ciphers of 3D point clouds. To address these challenges, Jolfaei et al. (2015) proposed a 3D object encryption algorithm along with geometric notions of dimensional and spatial stability. However, these notions are not consistent, and the geometric stability and correctness of that algorithm are not guaranteed as we show through counterexamples. In this paper, we introduce two enhanced ciphers with correctness, reversibility, and geometric stability guarantees. These ciphers employ chaotic permutations with hyperchaotic maps of highly complex behavior for enhanced security. The permutation scheme ensures the creation of consistent solvable equations for the decryption stage. Also, an enhanced 3D point rotation scheme is proposed to ensure geometric stability. The soundness and significance of the proposed ciphers are demonstrated by rigorous mathematical proofs. As well, extensive experimentation and comparisons against state-of-the-art methods are demonstrated through similarity analysis based on the Hausdorff and Euclidean distances, analysis of the sensitivity to plaintext and key perturbations, and analysis of the robustness to statistical attacks.
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We use a new version of Kramer's theorem to derive a sampling theorem associated with second order boundary-value problems whose eigenvalues are not necessarily simple.
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Integral transforms like Laplace, Mellin, and Fourier transforms are used in finding explicit solutions for linear differential equations, linear fractional differential equations, and diffusion ...equations. See for example (Kilbas et al., Theory and Applications of Fractional Differential Equations, Elsevier, London, first edition, 2006; Mainardi, Appl. Math. Lett. 9(6), 23–28, 1996; Nikolova and Boyadjiev, Fract. Calc. Appl. Anal. 13(1), 57–67, 2010; Wyss, J. Math. Phys. 27(11), 2782–2785, 1986). This chapter is devoted to the use of the q-Laplace, q-Mellin, and q2-Fourier transforms to find explicit solutions of certain linear q-difference equations, linear fractional q-difference equations, and certain fractional q-diffusion equations.
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We use a discrete version of Kramer's sampling theorem to derive sampling expansions for discrete transforms whose kernels arise from regular difference equations. The kernels may be taken to be ...either solutions or Green's functions of second order regular self-adjoint boundary-value problems. In both cases, the sampling eexpansions obtained are written in forms of finite-Lagrange-type interpolation expansions. Cauchy-type sampling expansions for periodic bandlimited signals will be derived using first order boundary-value problems. Illustrative exampls are also given.
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It is known that Kramer's sampling theorem and Lagrange-type interpolation generalize the celebrated Whittaker-Shannon-Kotel'nikov sampling theorem in two different directions; however, no direct ...connection between these two directions seems to be known. In this article we show that Kramer's sampling theorem gives nothing more than the Lagrange-type interpolations provided that the kernel function associated with Kramer's theorem arises from a self-adjoint boundary value problem with sample eigenvalues.
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We give sampling theorems associated with boundary value problems whose differential equations are of the form M(y) = λS(y), where M and S are differential expressions of the second and first order ...respectively and the eigenvalue parameter may appear in the boundary conditions. The class of the sampled functions is not a class of integral transforms as is the case in the classical sampling theory, but it is a class of integrodifferential transforms. We use solutions of the problem as well as Green's function to derive two sampling theorems.
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30.
Preliminaries Annaby, Mahmoud H.; Mansour, Zeinab S.
q -Fractional Calculus and Equations,
07/2012
Book Chapter
This chapter includes definitions and properties of Jackson q-difference and q-integral operators, q-gamma and q-beta functions and finally q-analogues of Laplace and Mellin integral transforms.
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