Linear Equation System Study on Electrical Circuits Using Matlab

The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

The nonlinear dynamics model of gearing system is developed based on RV transmission system. The influence of the nonlinear factors as time-varying meshing stiffness, backlash of the gear pairs and errors is considered. By means of the Lagrange equation the multi-degree-of-freedom differential equations of motion are derived. The differential equations are very hard to solve for which are characterized by positive semi-definition, time-variation and backlash-type nonlinearity. And linear and nonlinear restoring force are coexist in the equations. In order to solve easily, the differential equations are transformed to identical dimensionless nonlinear differential equations in matrix form. The establishment of the nonlinear differential equations laid a foundation for The Solution of differential equations and the analysis of the nonlinearity characteristics.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

The construction of composite models for the mechanics of a support of a set of rigid bodies

An effective fitting scheme within the thermo-viscoelastic model is proposed for estimation of dynamic eigenmodes of an ionic melt from ab initio molecular dynamics (AIMD) simulations. Dynamic eigenmodes are solutions of the generalized Langevin equation in matrix form, obtained on the basis set of eight dynamic variables. Fitting parameters are used only in the generalized hydrodynamic matrix T(k) for matrix elements involving heat density and heat current correlations only, because of huge computational efforts needed to calculate them directly in ab initio simulations. In the proposed scheme six AIMD-derived time correlation functions, three partial density-density and three partial current-current ones, are recovered by the proposed theoretical approach, which satisfies the exact sum rules up to the fourth frequency moments of partial dynamic structure factors. The suggested theoretical analysis was applied to estimation of propagating hydrodynamic and non-hydrodynamic eigenmodes in molten NaCl.

Double-shifted Chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral, Volterra integral equation, and Fredholm integral equation. This method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form. In addition, the computational complexity can be reduced remarkably. Three examples are illustrated. It is seen that the proposed approach is straightforward and convenient, and converges faster in finding approximations than other existing orthogonal function methods.

Chapter 2 - Simultaneous Linear Equations

A system of linear interval equations is called solvable if each system of linear equations contained therein is solvable. In the main result of this paper it is proved that solvability of a general rectangular system of linear interval equations can be characterized in terms of nonnegative solvability of a finite number of systems of linear equations which, however, is exponential in matrix size; the problem is proved to be NP-hard. It is shown that three earlier published results are consequences of the main theorem, which is compared with its counterpart valid for linear interval inequalities that turn out to be much less difficult to solve.

Formulas are derived for the constants ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ of superconducting alloys for arbitrary temperatures and impurity concentrations. Beginning with Gorkov's equations in matrix form, we calculate the free-energy density up to fourth order in $|\ensuremath{\Delta}|$, using Abrikosov's assumption of a fluxoid lattice. The impurities are treated by the usual averaging technique, retaining only $s$ and $p$ scattering, and a special technique is developed to treat exactly the nonvanishing commutator of different components of the gauge-invariant derivative. The results contain the already known limiting cases. Intermediate values are obtained by performing machine calculations using the general formulas developed here. It is found that the values of ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ drop relatively strongly even for small impurity concentration, and also depend unexpectedly strongly on the ratio of $s$ and $p$ scattering. The recent result of Caroli, Cyrot, and de Gennes is confirmed according to which ${\ensuremath{\kappa}}_{2}$ is approximately equal to ${\ensuremath{\kappa}}_{1}$ in the dirty limit.

The explicit finite-difference scheme is widely used in seismic wave simulation. Using large time steps can significantly reduce the number of iterations and thus improve computational efficiency, but this approach faces two main challenges: reduced accuracy caused by temporal dispersion and restrictions imposed by the Courant-Friedrichs-Lewy (CFL) stability condition. Based on the wavefield iteration equation in matrix form, an explicit finite-difference method with time step n-tupling and an extended CFL limit is developed for acoustic wave simulation. By combining n successive iteration operators into a single-step operator, two types of n-tupling algorithms are constructed, effectively expanding the CFL stability limit by a factor of n and enabling time steps well beyond conventional thresholds. Both theoretical analysis and numerical experiments demonstrate that simulations with time step n-tupling achieve accuracy equivalent to conventional single-step methods while reducing computation time to approximately 1/n, resulting in a corresponding n-fold increase in overall computational efficiency. Time dispersion is suppressed using two complementary strategies: for relatively small base time steps, n-tupling inherently attains the accuracy of smaller steps; for relatively large base steps, a time-dispersion transform is applied to eliminate errors and maintain high numerical accuracy throughout the simulation.

This paper proposes an algorithm for the analytical (with a controllable error) solution of boundary-value problems of shell deformation mechanics. Decomposition of partial differential equations is per-formed. The obtained ordinary differential equations in matrix form are solved in the form of converging matrix series in the corresponding coordinates. Atthe point of intersection of the coordinates of ordinary differential equations, the algebraic equations of their connection are satisfied. The error control of the solution of the boundary value problem is carried out by comparing the results when the grid of the de-composition of equations is refined. The study solved the problems of deformation a hinged cylindrical shell and a shallow shell fixed on the edges loaded with external pressure.

The detection performance of the matched filter with the transmitted signal as a reference signal will be degraded in an underwater acoustic channel due to the multipath effect and time variability. Two methods of the channel match are presented in this paper: (1) Pulse-to-pulse Correlation method; (2) Deconvolution method. In the first method the received signal is used as a reference signal of the matched filter. In the second method a short pulse is used to excite the channel. To obtain the response function of the channel, the Wiener-Hopf equation in matrix form is used. The solved response function of the channel is convoluted with the transmitted signal and the result of the convolution is used as a reference signal of the corrected matched filter. In this paper, with the aid of computer the corrected match for channel model and for the real underwater acoustic propagation channel is given.

The dynamical equations of a rotating body whose motion is referred to an arbitrarily rotating reference frame are often desired in linearized form in connection with stability and orientation control studies. The method of this paper shows how to obtain these linearized equations in matrix form, saving considerable labor and promoting accuracy relative to the linearization process done at the sealer level. Illustrative applications are made to a librating orbiting body and a one-axis gyroscope on a rotating base.

The physical model of semi-active quarter-coach suspension was established based on the theories of Buckingham’s Pi theorem in this paper. According to the characteristics of the semi-active suspension, a new intelligent control algorithm based on the Fuzzy logic control and cerebellar model articulation control techniques-Fuzzy cerebellar model articulation control (FCMAC) was presented and used to perform online control of semi-active suspension for the first time, novel weight-update laws were derived that guarantee the stability of closed-loop system, both information retrieval and learning rules were described by algebraic equations in matrix form. The results of experiment of closed -loop excited by three typical vibration signals showed that FCMAC control strategy can reduce the value of mean square root of vertical acceleration for semi-active suspension system to be lower 23%, 25% and 18% compared with the traditional control strategy such as PID.