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Integral and Integro-Differential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
One major application of the Mellin transform is of course to solve differential equations. Thus, we consider the following formula for taking the Mellin transform of the derivative of function f: M[tkdkf(t)dtk]=(-1)k(s)kF(s) $$ M[t^{k} \frac{{d^{k} f(t)}}{{dt^{k} }}] = ( - 1)^{k} (s)_{k} F(s) $$
You can create your own bifurcation
Published in International Journal of Mathematical Education in Science and Technology, 2021
Scientists are proud when their names are associated with a constant, a mathematical expression, or a theory. There are known shortcuts to trying to achieve this goal. For instance, this can be done by proposing a new kernel in an integral transform; that is, by properly choosing in , in order to map the function into the function . In the Fourier transform, ; in the Laplace transform, ; in the Mellin transform, (Polyanin & Chernoutsan, 2011). The main difficulty is to suggest a novel kernel that can be successfully used to solve practical problems. Another way to try to promote your name is to find a particularly special value of x around which a function can be written as a Taylor series expansion. The case x = 0 became known as Maclaurin series (Polyanin &Chernoutsan, 2011).
Optical colour image encryption using spiral phase transform and chaotic pixel scrambling
Published in Journal of Modern Optics, 2019
In the past few years, optical techniques for information security have attracted wide attention due to features such as parallel processing, fast computing and multidimensional capabilities for information processing (1). Among them, the double random phase encoding (DRPE) is the most studied technique. In DRPE, an input image is encoded into a stationary white noise using two random phase masks placed at the input and the Fourier planes, respectively (2). With time different encryption domains, such as Fresnel transform (FrT) (3, 4), fractional Fourier transform (FrFT) (5), gyrator transform (6), Mellin transform (7), have been developed for enhanced security. However, due to the inherent linearity in the DRPE architecture, these schemes were found vulnerable to various attacks (8, 9). To address the linearity issue, some asymmetric cryptosystems based on phase-truncated Fourier transform (PTFT) have also been proposed (10, 11). However, an iterative phase retrieval algorithm based special attack allows an attacker to reveal the encrypted information using these cryptosystems (12). An improved DRPE based technique using Kronecker product and hybrid phase masks is also introduced recently which is robust against the Known-plaintext attack (13) and enhances the security.
Asymmetric hybrid encryption scheme based on modified equal modulus decomposition in hybrid multi-resolution wavelet domain
Published in Journal of Modern Optics, 2019
Pankaj Rakheja, Rekha Vig, Phool Singh
The optical cryptosystems have gained a lot of popularity among researchers due to their inherent properties of large information capacity, parallel processing and high speed. Refregier and Javidi (1) proposed double random phase encoding (DPRE) in 1995. DPRE-based optical schemes were investigated and further enhanced by various researchers using different techniques: in fractional Fourier domain (2, 3), in Fresnel domain (4), using amplitude modulation (5), using fractional Fourier transform in digital holography (6), using diffractive imaging (7), using phase retrieval algorithm and intermodulation in Fourier domain (8). To improve the security of optical encryption schemes, various advanced technologies are combined in many ways such as image encoding based on multi-stage and multi-channel fractional Fourier transform (9), random binary phase modulation with mixture retrieval type of Yang-Gu algorithm (10), gyrator and Arnold transform (11), digital holography and joint correlators (12, 13), fractional Mellin transform (14), Hartley transform (15), Arnold transform and singular value decomposition in fractional Hartley domain (16), wavelet domain (17), gyrator wavelet transform (18), phase shifting interferometry (19), phase retrieval algorithm (20), photo counting and photo counting polarimetric image encryption (21, 22), compression-based image encryption (23, 24), diffraction imaging-based encryption and their vulnerability to ptychographic phase retrieval (25), encryption based on computational ghost imaging (26), encryption based on quantum imaging (27) and so on.