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In this paper the authors study the class of locally graded groups all of whose subgroups are permutable or are soluble and satisfy a certain rank condition. The rank conditions in question include groups of finite abelian section rank, minimax groups and polycyclic groups. In each case necessary and sufficient conditions are given for a locally graded group to have all subgroups permutable or soluble with the given rank condition.

Ecosystems can be abstracted into models consisting of compartments containing matter or energy, transactional flows of matter or energy between compartments, inputs into the system, and outputs from the system. Although direct transactions are measurable in the field, indirect transactions have been demonstrated to have dominant effects. Integral network utility (U) is a summation of all direct and indirect net transactions in a network presented in matrix format and developed as a feature of Network Environ Analysis (NEA). While U can provide qualitative information about ecological interactions between compartments, the nonzero-sum nature of indirect net transactions has made ecological interpretation of quantitative network utility challenging. Here we aimed to examine U for nine 2- or 3-compartment ecosystem models from a throughflow perspective. For each model, we assigned inputs, outputs, and flows algebraically using flow components traceable across the model, developed corresponding flow (F) and throughflow (T) matrices based on these values, and used symbolic Matlab to calculate the net adjacent flow intensity matrix (D) and U. Substituting algebraic combinations of flow components with corresponding throughflow values allowed us to reduce elements of U to throughflows to the maximum extent possible. Models with only simple input environs were fully throughflow reducible, while models with more complex input environs exhibited one to three nonreducible elements in U. Throughflow reducibility was sufficient, but not necessary, for topological determination of ecological relations of a model, as described by sign(U). Parametrically determined elements of sign(U), along with the specific flow components influencing the sign of that element, could be readily identified based on quantitative consideration of nonreducible flow components. We provide an example showing that considering throughflow as a centrality measure can allow the identification of a quantitative basis for network synergism. By allowing identification of specific subsets of transactional flows relating to ecosystem complexity and qualitative differences between human-designed systems in the conventional industrial model and evolved ecological systems, the throughflow perspective of U opens avenues for designing more sustainable human systems.

In this article, we do the cryptanalysis of the Enhanced and Secured RSA Key Generation Scheme (ESRKGS) given in Thangavel etźal. (2015). The authors claimed that ESRKGS is a highly secure and not easily breakable scheme compared to the traditional RSA scheme. In their scheme, the public key n is a product of two large prime numbers, but the values of Encryption E and Decryption D keys are based on the product of four large prime numbers, N, which is a multiple of n. They claimed that even if someone factorizes n, in order to obtain the private key D, brute force must be used to obtain the other two primes.We prove that we can use an alternative private key D1 to break the system. Someone does not need to employ brute force to break the system when n is factored. Hence, ESRKGS has the same security level as the traditional RSA.