Finding Roots of System of Equations

2022 ◽  
pp. 229-239
Author(s):  
Haksun Li, PhD
Keyword(s):  
2020 ◽  
Vol 23 (1) ◽  
pp. 97-101
Author(s):  
Mikhail Petrichenko ◽  
Dmitry W. Serow

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.


Author(s):  
N. S. Aryaeva ◽  
E. V. Koptev-Dvornikov ◽  
D. A. Bychkov

A system of equations of thermobarometer for magnetite-silicate melt equilibrium was obtained by method of multidimensional statistics of 93 experimental data of a magnetite solubility in basaltic melts. Equations reproduce experimental data in a wide range of basalt compositions, temperatures and pressures with small errors. Verification of thermobarometers showed the maximum error in liquidus temperature reproducing does not exceed ±7 °C. The level of cumulative magnetite appearance in the vertical structure of Tsypringa, Kivakka, Burakovsky intrusions predicted with errors from ±10 to ±50 m.


Author(s):  
A. M. Oleynikov ◽  
L. N. Kanov

The paper gives the description of the original wind electrical installation with mechanical reduction in which the output of vertical axis wind turbine with rather low rotation speed over multiplicator is distributed to a certain number of generators. The number of acting generators is determined by the output of actual operating wind stream at each moment. According to this constructive scheme, it is possible to provide effective and with maximum efficiency installation work in a wide range of wind speeds and under any schedule issued to the consumer of electricity. As there are no any experience in using such complexes, mathematical description of its main elements is given, namely windwheels, generators with electromagnetic excitation of magnetic electrical type, then their interaction with windwheel, and also the results of mathematical modeling of work system regimes under using the offered system of equations. The basis for the mathematical description of the main elements of the installation – synchronous generators – are the system of equations of electrical and mechanical equilibrium in relative units in rotating coordinates without considering saturation of the magnetic circuit. The equation of mechanical equilibrium systems includes torque and brake windwheel electromagnetic moments of generators with taking into account the reduction coefficients and friction. In addition, we specify the alternator rotor dynamics resulting from continuous torque of windwheel fluctuations under the influence of unsteady wind flow and wind speed serving as the original variable is modeled by a set of sinusoids. Model simplification is achieved by equivalization of similar generators and by disregarding these transitions with a small time constant. Calculation the installation with synchronous generators of two types of small and medium capacity taking into account the operational factors allowed us to demonstrate the logic of interactions in the main elements of the reported complex in the process of converting wind flow into the generated active and reactive power. We have shown the possibility of stable system work under changeable wind stream condition by regulating of the plant blade angle and with simultaneous varying of generator number of different types. All these are in great interest for project organizations and power producers.


2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.


2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.


1978 ◽  
Vol 45 (4) ◽  
pp. 711-716 ◽  
Author(s):  
Stephen S.-H. Chang

This paper presents a method for computing the motion and decay of a large dusty, buoyant thermal (cloud) carried by a vortex ring generated from a strong near ground explosion and ascending in an inhomogeneous atmosphere. A system of equations is derived describing the motion of the vortex ring, the thermal, and the pollutants which consist of numerous solid spherical particles. The interior properties and the trajectories of the thermal and the pollutants are obtained. The numerical solution for the thermal trajectory is in excellent agreement with experiment.


1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.


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