ORGANIZATION OF STUDENTS 'PROJECT ACTIVITIES IN PHYSICS LESSONS

The article shows that in the design process students are required not only to identify but also to substantiate certain essential features of the content component of the school course of physics. Therefore, forming a holistic view of the components, you can limit yourself to a system of questions that play the role of cognitive tasks, solving which in small groups, students get one or more essential features of the component. The most rational ways of formation and use in the course of training of the generalized systems of independent works of the pupils directed on development of their creative activity, independence, thinking are specified.

The Solvability of Generalized Systems of Time-Dependent Hemivariational Inequalities Enjoying Symmetric Structure in Reflexive Banach Spaces

In real reflexive Banach spaces, let the GSTDHVI, SHVI, DVIP, VIT, and KKM represent a generalized system of time-dependent hemivariational inequalities, a system of hemivariational inequalities, a derived vector inclusion problem, Volterra integral term, and Knaster–Kuratowski–Mazurkiewicz, respectively, where the GSTDHVI consists of two parts which are of symmetric structure mutually. By virtue of the surjectivity theorem for pseudo-monotonicity mappings and the Banach contraction mapping principle, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, we consider and study a GSTDHVI with VITs. Under quite mild assumptions, it is shown that there exists only a solution to the investigated problem via demonstrating that a DVIP with VIT is solvable.

k-Steps Explicit Iterative Algorithms to Solve Generalized System of Nonlinear Mixed Variational Inequalities

The objective of this paper is to consider new generalized systems of nonlinear mixed variational inequalities including 3k-distinct nonlinear relaxed cocoercive operators. We proposed k-steps explicit iterative methods including projection operators for this considered system and present the equivalent fixed point problem of this new generalized systems of variational inequalities. This equivalent fixed point problem suggest to us k-steps explicit iterative algorithms to obtain an approximate solution of the considered system. Convergence results of k-step explicit iterative algorithms are obtained.

Mathematical modelling of vertical and longitudinal-angular vibrations in the composition of the rigidity and damping of the hierarchical parts of the tractor taking into account road impact

The article contains generalized systems of equations describing the vertical and longitudinal-angular vibrations of the main parts of the tractor with the mathematical model of the hierarchical vertical linear oscillation of the tractor, the irregularities of the roads, taking into account the rigidity and damping. It includes systems: equations describing the vertical vibrations of the skeleton over the propellers of a tractor on an uneven road; equations describing the vertical vibrations of the cab of the rear of the tractor, taking into account the walking of the tractor on uneven roads. Equation systems have also been compiled that describe the vertical oscillations of the uppermost hierarchical link - the seat of the tractor operator, taking into account the walking of the tractor on an uneven road. The use of this original mathematical model will make it possible to find random unwanted oscillations of the new designed machine design. It will help to optimize their geometric and dynamic parameters.

Multivaluedness in Networks: Theory

<div><div>This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded1 systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufficient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. </div><div><br></div><div>The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems.</div><div><br></div><div>The significance of the work presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.</div></div>

Multivaluedness in Networks: Theory

<div><div>This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded1 systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufficient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. </div><div><br></div><div>The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems.</div><div><br></div><div>The significance of the work presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.</div></div>

Three Types Generalized Zn-Heisenberg Ferromagnet Models

By taking values in a commutative subalgebra gln,C, we construct a new generalized Zn-Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized Zn-Heisenberg ferromagnet model and Zn-mixed derivative nonlinear Schrödinger equation has been investigated. The Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized Zn-inhomogeneous Heisenberg ferromagnet model and Zn-Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Zn-nonlinear Schrödinger equation, Zn-Davey–Stewartson equation and their Lax representation have been well studied.

Systems of k Boolean inequations and a Boolean equation

In this paper elementary generalized systems of Boolean equations are investigated. The formula for solving systems of k Boolean inequations and a Boolean equation is presented. This systems have many applications in computer science for solving logical problems. Presented formulas can accelerate application of elementary generalized systems of Boolean equations.