Classes of Dynamic Systems with Various Combinations of Multipliers in Their Reciprocal Polynomial Right Parts

A family of differential dynamic systems is considered on a real plane of their phase variables x, y. The main common feature of systems under consideration is: every particular system includes equations with polynomial right parts of the third order in one equation and of the second order in another one. These polynomials are mutually reciprocal, i.e., their decompositions into forms of lower orders do not contain common multipliers. The whole family of dynamic systems has been split into subfamilies according to the numbers of different reciprocal multipliers in the decompositions and depending on an order of sequence of different roots of polynomials. Every subfamily has been studied in a Poincare circle using Poincare mappings. A plan of the investigation for each selected subfamily of dynamic systems includes the following steps. We determine a list of singular points of systems of the fixed subfamily in a Poincare circle. For every singular point in the list, we use the notions of a saddle (S) and node (N) bundles of adjacent to this point semi trajectories, of a separatrix of the singular point, and of a topo dynamical type of the singular point (its TD – type). Further we split the family under consideration to subfamilies of different hierarchical levels with proper numbers. For every chosen subfamily we reveal topo dynamical types of singular points and separatrices of them. We investigate the behavior of separatrices for all singular points of systems belonging to the chosen subfamily. Very important are: a question of a uniqueness of a continuation of every given separatrix from a small neighborhood of a singular point to all the lengths of this separatrix, as well as a question of a mutual arrangement of all separatrices in a Poincare circle Ω. We answer these questions for all subfamilies of studied systems. The presented work is devoted to the original study. The main task of the work is to depict and describe all different in the topological meaning phase portraits in a Poincare circle, possible for the dynamical differential systems belonging to a broad family under consideration, and to its numerical subfamilies of different hierarchical levels. This is a theoretical work, but due to special research methods it may be useful for applied studies of dynamic systems with polynomial right parts. Author hopes that this work may be interesting and useful for researchers as well as for students and postgraduates. As a result, we describe and depict phase portraits of dynamic systems of a taken family and outline the criteria of every portrait appearance.

Moment Estimates of the Cloud of a Planar Measure

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.

Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied.

Extensions of power aggregation operators for decision making based on complex picture fuzzy knowledge

This article puts forward an innovative notion of complex picture fuzzy set (CPFS) which is particularly an extension and a generalization of picture fuzzy sets (PFSs) by the addition of phase term in the description of PFSs. The uniqueness of CPFS lies in the capability to manage the uncertainty and periodicity, simultaneously, due to the presence of phase term which broadens the range of CPFS from a real plane to the complex plane of unit disk. We describe and verify the fundamental operations and properties of CPFSs. We introduce the aggregation operators, namely; complex picture fuzzy power averaging and complex picture fuzzy power geometric operators in CPFSs environment, based on weighted and ordered weighted averaging and geometric operators. We construct multi-criteria decision making (MCDM) problem, using these operators and describe a numerical example to illustrate the validity and competence of this article. Finally, we discuss the advantages of this generalized concept of aggregation technique and analyze a comparative study to demonstrate the superiority and consistency of our model.

Level lines of a polynomial in the plane

We propose a method for computing the position of all level lines of a real polynomial in the real plane. To do this, it is necessary to compute its critical points and critical curves, and then to compute critical values of the polynomial (there are finite number of them). Now finite number of critical levels and one representative of noncritical level corresponding to a value between two neighboring critical ones enough to compute. We propose a scheme for computing level lines based on polynomial computer algebra algorithms: Gröbner bases, primary ideal decomposition. Software for these computations are pointed out. Nontrivial examples are considered.

Examples of computation of level lines of polynomials in a plane

Here we present a theory and 3 nontrivial examples of level lines calculating of real polynomials in the real plane. For this case we implement the following programs of computational algebra: factorization of a polynomial, calculation of the Grobner basis, construction of Newton's polygon, representation of an algebraic curve in a plane. Furthermore, it is shown how to overcome computational difficulties.

Directional launching of surface plasmon polaritons by electrically driven aperiodic groove array reflectors

Access to surface plasmon polaritons (SPPs) with directional control excited by electrical means is important for applications in (on-chip) nano-optoelectronic devices and to circumvent limitations inherent to approaches where SPPs are excited by optical means (e.g., diffraction limit). This paper describes directional excitation of surface plasmon polaritons propagating along a plasmonic strip waveguide integrated with an aperiodic groove array electrically driven by an Al–Al2O3–Au tunnel junction. The aperiodic groove array consists of six grooves and is optimized to specifically reflect the SPPs by 180° in the desired direction (+x or −x) along the plasmonic strip waveguide. We used constrained nonlinear optimization of the groove array based on the sequential quadratic programming algorithms coupled with finite-difference time-domain (FDTD) simulations to achieve the optimal structures. Leakage radiation microscopy (Fourier and real plane imaging) shows that the propagation direction of selectively only one SPP mode (propagating along the metal–substrate interface) is controlled. In our experiments, we achieved a directionality (i.e., +x/−x ratio) of close to 8, and all of our experimental findings are supported by detailed theoretical simulations.

Dulac – Cherkas functions for systems equivalent to the van der Pol equation

The object of this study is an autonomous van der Pol system on a real plane. The subject of the study is the properties of the limit cycle of this system. The main purpose of this paper is to find the localization of the limit cycle on the phase plane and establish its shape for various values of the real parameter of the van der Pol system. Our approach is based on the use of transverse curves related to the Dulac – Cherkas functions and approximating the location of the limit cycle. As the first step, five topologically equivalent systems, including systems with a parameter rotating the vector field, as well as singularly perturbed systems are determined for the van der Pol system. Then, applying the previously elaborated method, we constructed two polynomial Dulac – Cherkas functions for each of three systems from the considered ones in the phase plane for all real nonzero values of the parameter. Using them, transverse curves forming the boundaries of the localization regions of the limit cycle for the van der Pol system are found. Thus, the constructed Dulac – Cherkas functions allow us to determine the location of the limit cycle on the basis of algebraic curves for all real parameter values, including values close to the bifurcation of a limit cycle from the center ovals, the Andronov – Hopf bifurcation, and the bifurcation from a closed trajectory related to a discontinuous periodic solution.

On the Topological Structure and Properties of Multidimensional (C, R) Space

Generally, the linear topological spaces successfully generate Tychonoff product topology in lower dimensions. This paper proposes the construction and analysis of a multidimensional topological space based on the Cartesian product of complex and real spaces in continua. The geometry of the resulting space includes a real plane with planar rotational symmetry. The basis of topological space contains cylindrical open sets. The projection of a cylindrically symmetric continuous function in the topological space onto a complex planar subspace maintains surjectivity. The proposed construction shows that there are two projective topological subspaces admitting non-uniform scaling, where the complex subspace scales at a higher order than the real subspace generating a quasinormed space. Furthermore, the space can be equipped with commutative and finite translations on complex and real subspaces. The complex subspace containing the origin of real subspace supports associativity under finite translation and multiplication operations in a combination. The analysis of the formation of a multidimensional topological group in the space requires first-order translation in complex subspace, where the identity element is located on real plane in the space. Moreover, the complex translation of identity element is restricted within the corresponding real plane. The topological projections support additive group structures in real one-dimensional as well as two-dimensional complex subspaces. Furthermore, a multiplicative group is formed in the real projective space. The topological properties, such as the compactness and homeomorphism of subspaces under various combinations of projections and translations, are analyzed. It is considered that the complex subspace is holomorphic in nature.