Hypermatrix factors for string and membrane junctions

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provide such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of three-junctions of strings and membranes. Generalizations are also considered.Comment: 25 pages, 4 figures, presentation improved, minor correction

Estimation of Design Parameters of the Crank-Connecting Rod Mechanism of Engines for Mobile Agricultural Machines

An accurate choice of the design model of the crank-connecting rod mechanism of piston internal combustion engines affects the accuracy of the calculation results and their complexity. At present, most of scientists and technicians choose a two-mass design model to analyze the operation of the crankconnecting rod mechanism. The model considers only the rotational and reciprocating movements of two masses, which are connected by a rigid weightless rod. This model significantly simplifies the calculations, neglects the elastic deformations of the parts of the crank-connecting rod mechanism, and eliminates the need for compiling the equations of dynamics in partial derivatives. However, the model has a number of drawbacks. The calculation results obtained using the two-mass model exhibit significant errors, which mainly depend on the design features of the connecting rod assembly. The paper discusses multi-mass design models, where the connecting rod assembly can comprise several lumped masses located along its length. In this case, the plane-parallel motion of these masses is added. The masses have weightless and absolutely rigid bonds. Forces and moments acting on the piston assembly and the crank are calculated according to the equations compiled. Comparison of the calculation results with the results obtained for a two-mass model can be used to determine errors and choose a design model that provides the required accuracy. The considered design model is of interest to engineers and technicians engaged in the design and calculation of the crank-connecting rod mechanism of piston internal combustion engines

Justification of the optimal annual load on the tractor providing for its parameters stress on the formed crop

The article presents the results of a study to identify the optimal value of the annual load on one tractor using direct sowing technology. The optimal annual load on one tractor is directly associated with the power provision to agriculture and, indirectly, with its productivity. The article analyzes the dependence of the annual tractor load in ploughland per ha on various factors of the tractor-operator-instrument-fieldsoil-crop system: on tractor parameters; unit; environment and it is concluded that the optimal annual load on one tractor depends on them. The method of choosing various agricultural enterprises power supply should be connected with the natural and climatic conditions of their production activities, accepted technologies for cultivating crops, soil types, field sizes, etc. It was revealed that in Russia the optimal energy supply of agriculture with the power of tractor engines, using the zero tillage technology, is in the range from 2.4 to 2.8 hp/ha

Energy Justification of the Number of Tractors for Agricultural Operations

The relevance of the problem under study is due to the need to substantiate the sufficient number of tractors used for various technological operations, depending on the volume of works performed, and taking into account their influence on the crop yield. The purpose is to develop the mathematical model of machine and tractor units used in technological operations by formalizing the vague concept of optimal agrotechnical timing of technological operations and calculating the optimal number of tractors to perform works by the total energy costs. The main research method is mathematical modeling of operations performed by machine-tractor units based on the system analysis and computational experiments and using the energy mathematical model. The article describes a criterion for optimizing the tractor parameters, dependences of the energy mathematical model of the unit. The results of computational experiments showed that for each operation there is an optimal number of tractors and tractor units minimizing the minimum total energy costs. Calculations were carried out when treating 1000 hectares with various tractors. It was revealed that the total energy consumption differs by 2 or more times. The required number of tractors depends on their main parameters, propeller parameters and other factors. It was revealed that with an increase in the volume of works, the required optimal number of tractors increases. The method for substantiating the required optimal number of tractors in technological operations will improve operations of tractors used by large agricultural producers, reduce the total energy costs for cultivating up to 12–16 thousand MJ/ha

The moduli space of three qutrit states

19 pages, revtex4, three figures; revised versionInternational audienceWe study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behaviour of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC). We describe the orbit space of the SLOCC group SL(3,\C)^{\times 3} both in its affine and projective versions in terms of a very symmetric normal form parameterized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes