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It is shown how a stability test, alternative to the classical Routh test, can profitably be applied to check the presence of polynomial roots inside half-planes or even sectors of the complex plane. This result is obtained by exploiting the peculiar symmetries of the root locus in which the basic recursion of the test can be embedded. As is expected, the suggested approach proves useful for testing the stability of fractional-order systems. A pair of examples show how the method operates. It is believed that the suggested geometric approach can also be of some didactic value in introducing basic control-system tools to engineering students.
The geometric model accuracy is crucial for product design. More complex surfaces are represented by the approximation methods. On the contrary, the approximation methods reduce the design quality. A new alternative calculation method is proposed. The new method can calculate both conical sections and more complex curves. The researcher is able to get an analytical solution and not a sequence of points with the destruction of the object semantics. The new method is based on permutation and other symmetries and should have an origin in the internal properties of the space. The classical method consists of finding transformation parameters for symmetrical conic profiles, however a new procedure for parameters of linear transformations determination was acquired by another method. The main steps of the new method are theoretically presented in the paper. Since a double result is obtained in most stages, the new calculation method is easy to verify. Geometric modeling in the AutoCAD environment is shown briefly. The new calculation method can be used for most complex curves and linear transformations. Theoretical and practical researches are required additionally.
In this paper we extend the method of obtaining symmetries of ordinary differential equations to first order non-homogeneous neutral differential equations with variable coefficients. The existing method for delay differential equations uses a Lie-Bäcklund operator and an Invariant Manifold Theorem to define the operators which are used to obtain the infinitesimal generators of the Lie group. In this paper, we adopt a different approach and use Taylor’s theorem to obtain a Lie type invariance condition and the determining equations for a neutral differential equation. We then split this equation in a manner similar to that of ordinary differential equations to obtain an over-determined system of partial differential equations. These equations are then solved to obtain corresponding infinitesimals, and hence desired equivalent symmetries. We then obtain the symmetry algebra admitted by this neutral differential equation.
Nonlocally related systems for the Euler and Lagrange systems of twodimensional dynamical nonlinear elasticity are constructed. Using the continuity equation, i.e., conservation of mass of the Euler system to represent the density and Eulerian velocity components as the curl of a potential vector, one obtains the Euler potential system that is nonlocally related to the Euler system. It is shown that the Euler potential system also serves as a potential system of the Lagrange system. As a consequence, a direct connection is established between the Euler and Lagrange systems within a tree of nonlocally related systems. This extends the known situation for one-dimensional dynamical nonlinear elasticity to two spatial dimensions.
Content available remote Some Remarks on Kappa-deformed Boost Transformations
We discuss the invariance of the action of the Lorentz algebra on the relativistic fourmomentum algebra under nonlinear transformations. In particular, we describe the global Lorentz boosts acting on four momenta in cases for classical and �Č-deformed Poincare relativistic symmetry.
Content available remote Finding symmetries of algebraic system nets
The problem of finding symmetry information from algebraic system nets prior to the reach-ability graph generation is studied. The approach presented is based on wellformedness of transition descriptions, meaning that some data types in a net may be used in a symmetric way. Permutations on the domains of such data types produce symmetries on the state space level of the net, which in turn can be exploited during the reachability analysis. To ensure that the transitions behave symmetrically with respect to the chosen data domain permutations, a sufficient compatibility condition between data domain permutations and the algebraic terms used as transition guards and arc annotations is proposed. The solution is a general and flexi-ble one as it does not fix the set of applicable operations, enabling the design of customized net classes. To help the process of deciding whether a term is compatible with a data domain permutation, an approximation rule for the compatibility condition is given.
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