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Asymmetric Cryptography
Published in Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Emerging Security Algorithms and Techniques, 2019
Rajiv Ranjan, Abir Mukherjee, Pankaj Rai, Khaleel Ahmad
A semigroup is a set G having a binary operation ● and satisfying only the group axioms of closure and associativity. A group having a finite number of elements is called a finite group; otherwise, it is called an infinite group. A group is said to have the order n, denoted |G| = n, if the number of elements in the group is n. A nonempty subset of G, satisfying all the group axioms under the same binary operation, is called a subgroup of G.
Aerosol diffusion battery: The retrieval of particle size distribution with the help of analytical formulas
Published in Aerosol Science and Technology, 2018
A. A. Onischuk, A. M. Baklanov, S. V. Valiulin, P. P. Moiseenko, V. G. Mitrochenko
For numerical computation, the particle size distribution function is most often represented in the finite-dimensional form. A variety of numerical methods had been published to solve the inverse problem Equation (3). These methods include linear and nonlinear iterative techniques (Twomey 1975; Reineking and Porstendörfer 1986; Ferri et al. 1995), Tihonov's regularization procedure (Bashurova et al. 1992; Wang and Yangm 2008), least square methods (Voutilainen et al. 2001; Fierz et al. 2008), expectation-maximization algorithm (Maher and Laird 1985), matrix factorization methods (Paatero et al. 1991), the maximum entropy method (Yee 1989; Eremenko and Ankilov 1995; Gulak et al. 2010), the lognormal simplex method (Reineking and Porstendörfer 1986), extreme value estimation method (Aalto et al 1990) and others. The major drawbacks of numerical methods are the non-unicity of the solution and mathematically complex procedure to find the best solution out of the infinite group of size distributions giving penetrations that coincide within the experimental uncertainty with the measured ones.
On infinite group velocity of Lamb waves
Published in Waves in Random and Complex Media, 2023
Herein, a geometric condition for the infinite group velocity (IGV) is introduced, revealing that this condition splits into several conditions depending on phase velocity and frequency. One of these conditions is applied to searching zones with high group velocity at Lamb wave propagation in homogeneous anisotropic crystals. It is shown that according to Biot’s relation between elastic energy and the group velocity [10], the IGV condition means either vanishing of strain energy or/and infinite increase of the corresponding energy flux. Various applications in the nondestructive testing of materials and structures by emission of the guided acoustic waves [44], can be anticipated.
Equivariant perturbation in Gomory and Johnson's infinite group problem (V). Software for the continuous and discontinuous 1-row case
Published in Optimization Methods and Software, 2018
Chun Yu Hong, Matthias Köppe, Yuan Zhou
Another important way to obtain a discrete function is to restrict a piecewise linear function π for the infinite group problem to the cyclic group of given order q, by calling the procedure restrict_to_finite_group. Conversely, the procedure interpolate_to_infinite_group provides the piecewise linear interpolation π of a given discrete function .