AUT Journal of Mathematics and Computing
Volume:3 Issue: 1, Feb 2022

In the present work, we study the optimal control paths in the Zermelo navigation problem from the geometric and differential equations point of view rather than the optimal control point of view, where the latter has been carried out in our recent work. Here, we obtain the precise form of the system of ODE where the solutions are optimal trajectories of Zermelo's navigation problem. Having a precise equation allows optimizing a cost function more accurately and efficiently. The advantage of these equations is to approximate optimal trajectories in the general case by the first order approximation of external fields ω. The latter could be solved numerically since we have retrieved simpler equations for these paths.

‎The a-number is an invariant of the isomorphism class of the p-torsion group scheme‎. ‎In this paper‎, ‎we compute a closed formula for the a-number of yq + ‎y =x(q+1)/3 and  $sum_{t=1}^{s} y^{q⁄ 3^t}= xq+1$ with q = 3s over the finite field‎ ‎Fq2 using the action of the Cartier operator on H0(C,Ω1).

In this paper group-invariant properties of the Dym equation are studied. Lie symmetries are given and some group-invariant solutions are found with the use of similarity variables obtained from these operators. Conservation laws are computed via three methods. Direct method for construction of conservation laws is introduced by the concept of multipliers and Euler-Lagrange operator. Next, the non-linearly self-adjointness of the considered PDE is stated. Then, the modified Noether's theorem is used for finding conservation laws. Finally, the third method is established via the Hereman-Pole method by using the evolutionary form of the equation.

In this paper, we consider the generalized Kropina conformal change of m-th root metric and for this, prove a necessary and sufficient condition of locally projectively flatness. Also we proved a necessary and sufficient condition for the generalized Kropina conformal change of m-th root metric is locally dually flat.

The prime graph (or Gruenberg-Kegel graph) of a finite group is a well-known graph. In this paper, first, we investigate the structure of the finite groups with a non-complete prime graph. Then as an application, we prove that every alternating group An, where n≤31 is determined by its order and its largest element order. Also, we show that A32 is not characterizable by order and the largest element order.

A collection μ of subsets of a nonempty set X is a supra topology on X whenever ∅ and X belong to μ, and also μ isclosed under arbitrary unions. Also, a nonempty collection S of nonempty subsets of a nonempty set X is called a stack on X whenever it is closed under operation superset. In this paper, we are going to introduce an approach to extract some supra topologies from the topology of a topological space. For this purpose, we consider a topological space (X, τ) with a closed set P of its subsets. Using a stack S on the space (X, τ) and the closure operator cl associated with τ, we define a supra closure operator λP on X to create the desired supra topology. We then characterize the form of this resulting supra topology and also determine its relationship to the initial topology of the space.

It is well known that, one of the useful and rapid methods for a nonlinear system of algebraic equations is Newton's method. Newton's method has at least quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. In this paper, a differential continuation method is presented for solving the nonlinear system of algebraic equations whose Jacobian matrix is singular at the solution. For this purpose, at first, an auxiliary equation named the homotopy equation is constructed. Then, by differentiating from the homotopy equation, a system of differential equations is replaced instead of the target problem and solved. In other words, the solution of the nonlinear system of algebraic equations with singular Jacobian is transformed to the solution of a system of differential equations. Some numerical tests are presented at the end and the computational efficiency of the method is described.

Essential genes and proteins as their products encode the basic functions of a cell in a variety of conditions and are vital for the survival of a cell. Analyzing the characteristics of these proteins provides important biological information. An interesting analysis is to demonstrate the correlation between topological importance of a protein in protein-protein interaction networks and its essentiality. Different centrality criteria such as degree, betweenness, closeness, and eigenvector centralities are used to investigate such a correlation. Despite the remarkable results obtained by these methods, it is shown that the centrality criteria in scale-free networks show a high level of correlations which indicate that they share similar topological information of the networks. In this paper, we use a different approach for analyzing this correlation and use a well-known problem in the field of graph theory, Critical Node Detection Problem and solve it on the protein-protein interaction networks to obtain a subset of proteins called critical nodes which have the most effect on the network stability. Our results show that essential proteins have a more prominent presence in the set of critical nodes than what expected at random samples. Furthermore, the essential proteins represented in the set of critical nodes have a different distribution of topological properties compared to the essential proteins recovered by the centrality-based methods.All the source codes and data are available at “http://bioinformatics.aut.ac.ir/CNDP_PPI_networks/”.

Nolan and Ojeda-Revah (2013) proposed a regression model with heavy-tailed stable errors. In this paper we extend this method for multivariate heavy-tailed errors. Furthermore, A likelihood ratio test (LRT) for testing significant of regression coefficients is proposed. Also, confidence intervals based on Fisher information for Nolan and Ojeda-Revah (2013) method, called NOR, and LRT are computed and compared with well-known methods. At the end we provide some guidance for various error distributions in heavy-tailed cases.

In distributed computing, the uncertainty is the most challenging issue which is caused by the asynchrony of distributed entities' communication and many other reasons such as geographical scattering of distributed entities, their mobility, and etc. In this paper, IOTA, a DAG based Distributed ledger technology is used in order to cope with asynchronous communications and uncertainty. Moreover, IOTA private network is chosen to deal with other mentioned problem inside distributed computing. As case study, a system is presented which could be implemented inside Tehran Polytechnic university to bring computational power of computers with low resources together in order to solve many problems which can be solved in distributed computing manner.

In this paper, a computably definable predicate in metric structures is defined and characterized. Then, it is proved that every separable infinite-dimensional Hilbert structure in an effectively presented language is computable. Moreover, every definable predicate in these structures is computable.

VoIP is naturally an unreliable communication system. Thus, using the best VoIP steganographic systems, the accuracy of the hidden message is impaired as a result of the VoIP packet loss. There are many steganography and steganalysis researches that try to improve the robustness and accuracy of VoIP steganography methods. In addition to the fact that these works are done depend on a particular method, none of them have solved the problem of packet loss. Applying error correcting codes, prior to embedding, is a wellknown technique in telecommunication to improve robustness and to reconstruct Missing data. However, in the case of VoIP communication, a codeword entirely embedded in the packet may be lost due to the packet loss and therefore ECC techniques will not be capable of reconstructing the lost bits. In this paper, we design a novel scheme to increase the reliability of VoIP steganography systems. We emphasize that our proposed method, independent of the embedding and extracting algorithm, can be used in all VoIP steganography systems. After encoding the secret message to the codewords of n bits, we distribute these n bits into n successive RTP packets, in such a way that, losing one packet leads to missing only one bit of each codeword. Then, with the idea of telecommunication solutions in recovering lost data, when up to t of n packets are lost we can recover the secret message using a t-error correcting code C(n,k,d). Provided that the average of packet loss over the network is less than 1%, using a t-error correcting code C(n,k,d), the probability of losing hidden data, in each category of n-packets, Pe, is less than 10-2t. Hence, applying the t-error correcting codes with larger t, in the proposed scheme, results in more reliable steganographic systems.