Thin-wire radiating structures with double symmetry

The problems of electrodynamic analysis of thin-wire radiating structures with double symmetry are considered. New generalized integral representations of the electromagnetic field are obtained for the case of structures with single and double symmetries. Based on the obtained expressions, mathematical models of two - and four-way elliptical spiral antennas are constructed. It is shown that taking into account double symmetry in solving the internal electrodynamic problem leads to a set of independent Fredholm integral equations of the first kind written with respect to the distributions of normal current waves, which significantly simplifies the solution of the internal electrodynamic problem. Comparisons of current distributions along conductors, their input resistance dependences on the radius of the structure, and normalized radiation patterns for two- and four-way spiral emitters are presented.

On reversible maps and symmetric periodic points

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.

ASPHERICAL SYMMETRIES OF GRAPH LINKS

We will classify 2-component non-splittable graph links in S3, each of which is aspherically symmetric, i.e. the exterior admits a periodic homeomorphism which does not extend to a periodic homeomorphism of S3. They can be obtained from spherically symmetric graph links by one or two steps of specific types of iterated cabling. Some of them admit both orientation-preserving and orientation-reversing aspherical symmetries. We will study whether they admit Dehn surgeries which yield 3-manifolds with the induced double symmetries or not.

DOUBLE STRUCTURES AND DOUBLE SYMMETRIES FOR THE EINSTEIN–KALB–RAMOND THEORY

We further study the two-dimensional reduced Einstein–Kalb–Ramond (EKR) theory in the axisymmetric case by using the so-called double-complex function method. We find a doubleness symmetry of this theory and exploit it so that some double-complex d×d matrix Ernst-like potential can be constructed, and the associated equations of motion can be extended into a double-complex matrix Ernst-like form. Then we give a double symmetry group [Formula: see text] for the EKR theory and verify that its action can be realized concisely by a double-complex matrix, form generalization of the fractional linear transformation on the Ernst potential. These results demonstrate that the theory under consideration possesses more and richer symmetry structures. Moreover, as an application, we obtain an infinite chain of double-solutions of the EKR theory showing that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.

DOUBLE SYMMETRIES IN FIELD THEORIES

In this letter a concept of double symmetry is introduced, and its qualitative characteristics and rigorous definitions are given. We describe two ways of constructing the double-symmetric field theories and present an example demonstrating the high efficiency of one of them. Noting the existing double-symmetric theories we draw attention to a dual status of the group SU (2)L ⊗ SU (2)R as a secondary symmetry group, and in this connection we briefly discuss logically possible aspects of the P-violation in weak interactions.