Fermat`s Last Theorem Proof on Six Faces of a Wooden Cube

Fermat's Last Theorem has been proved on the basis of school Physics, Mathematics, analytical Geometry. The main conceptions of the proof one can write on a math toy in the form of a wooden cube for children. Six faces of the cube are enough to deliver the main ideas of proof.

Robust Finite-Time Tracking Control for Robotic Manipulators with Time Delay Estimation

In this study, a robust H∞ finite-time tracking controller is proposed for robotic manipulators based on time delay estimation. In this controller, there is no need to know the dynamics of robots, so it is quite simple. The high-gain observer is employed to estimate the joint velocities, which makes it much lower in cost. The theorem proof shows that the closed-loop system is finite-time stable and has a L2 gain that is less than or equal to γ, which shows high accuracy and strong robustness to estimation errors and external disturbances. Simulations on a two-link robot illustrate the effectiveness and advantages of the proposed controllers.

The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

The sufficient conditions for polystability of solutions of nonlinear systems of ordinary differential equations

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

Formal Verification Method for Configuration of Integrated Modular Avionics System Using MARTE

The configuration information of Integrated Modular Avionics (IMA) system includes almost all details of whole system architecture, which is used to configure the hardware interfaces, operating system, and interactions among applications to make an IMA system work correctly and reliably. It is very important to ensure the correctness and integrity of the configuration in the IMA system design phase. In this paper, we focus on modelling and verification of configuration information of IMA/ARINC653 system based on MARTE (Modelling and Analysis for Real-time and Embedded Systems). Firstly, we define semantic mapping from key concepts of configuration (such as modules, partitions, memory, process, and communications) to components of MARTE element and propose a method for model transformation between XML-formatted configuration information and MARTE models. Then we present a formal verification framework for ARINC653 system configuration based on theorem proof techniques, including construction of corresponding REAL theorems according to the semantics of those key components of configuration information and formal verification of theorems for the properties of IMA, such as time constraints, spatial isolation, and health monitoring. After that, a special issue of schedulability analysis of ARINC653 system is studied. We design a hierarchical scheduling strategy with consideration of characters of the ARINC653 system, and a scheduling analyzer MAST-2 is used to implement hierarchical schedule analysis. Lastly, we design a prototype tool, called Configuration Checker for ARINC653 (CC653), and two case studies show that the methods proposed in this paper are feasible and efficient.