Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction

Besides its usual interpretation as a system of $n$ indistinguishable particles moving on the circle, the trigonometric Sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line, and these 3 physically distinct systems are in duality with corresponding variants of the rational Ruijsenaars-Schneider system. We explain that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the cotangent bundles of the group U(n) and its covering groups $U(1) \times SU(n)$ and ${\mathbb R}\times SU(n)$, respectively. This geometric interpretation enhances our understanding of the duality relations and simplifies Ruijsenaars' original direct arguments that led to their discovery.Comment: 34 pages, minor additions and corrections of typos in v

An infinitesimally nonrigid polyhedron with nonstationary volume in the Lobachevsky 3-space

We give an example of an infinitesimally nonrigid polyhedron in the Lobachevsky 3-space and construct an infinitesimal flex of that polyhedron such that the volume of the polyhedron isn't stationary under the flex.Comment: 10 pages, 2 Postscript figure

The quantitative analysis of trade policy: a strategy in global competitive conflict

Abstract: The quantitative analysis of trading wars is now an actual and modern problem although the first works on optimisation of the custom duties date back to the end of the 19th century. The problem of reducing the number of possible solutions to the conflict and precise identifying of the optimal strategies is important. Applying mathematics makes the problem more formal, allows removing political tension in search of optimum solutions to all participants of the trading conflict. We consider that some state sells a product manufactured both in the country and abroad to other countries. The country-consumer of the product faces a choice of the import regulation in relation to the situation by means of the import duty size. A goal of such regulation may be the income of taxes and customs payments or the growth of the domestic manufacturer part in sales volumes. A goal of the importer which it achieves by the import volume variation is either profit, or a number of workplaces in the countries making the imported product. We introduce three multicriteria problems which correspond to different purposes of economic agents. The Pareto set is constructed for each problem. The main attention is paid to the Pareto set reducing based on additional information on the preference relation of the economic agents. Several calculations are made and illustrative examples are given. Keywords: quantitative analysis; optimal tariff; trade policy; risk management; multicriteria problem; Pareto set

On the scattering theory of the classical hyperbolic C(n) Sutherland model

In this paper we study the scattering theory of the classical hyperbolic Sutherland model associated with the C(n) root system. We prove that for any values of the coupling constants the scattering map has a factorized form. As a byproduct of our analysis, we propose a Lax matrix for the rational C(n) Ruijsenaars-Schneider-van Diejen model with two independent coupling constants, thereby setting the stage to establish the duality between the hyperbolic C(n) Sutherland and the rational C(n) Ruijsenaars-Schneider-van Diejen models.Comment: 15 page

Calculations for the Practical Applications of Quadratic Helicity in MHD

For the quadratic helicity χ(2), we present a generalization of the Arnol'd inequality which relates the magnetic energy to the quadratic helicity, which poses a lower bound. We then introduce the quadratic helicity density using the classical magnetic helicity density and its derivatives along magnetic field lines. For practical purposes, we also compute the flow of the quadratic helicity and show that for an α2-dynamo setting, it coincides with the flow of the square of the classical helicity. We then show how the quadratic helicity can be extended to obtain an invariant even under compressible deformations. Finally, we conclude with the numerical computation of χ(2) which show cases the practical usage of this higher order topological invariant

A simplicial gauge theory

We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noether's theorem that can be applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a discrete Noether's theorem. v3: Section 4 on Noether's theorem has been expanded with Proposition 8, section 2 has been expanded with a paragraph on standard LGT. v4: Thorough revision with new introduction and more background materia

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

Determination of electromagnetic medium from the Fresnel surface

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by an antisymmetric $2\choose 2$-tensor $\kappa$. In this setting, the Tamm-Rubilar tensor density determines a polynomial surface of fourth order in each cotangent space. This surface is called the Fresnel surface and acts as a generalisation of the light-cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speed as a function of direction. Favaro and Bergamin have recently proven that if $\kappa$ has only a principal part and if the Fresnel surface of $\kappa$ coincides with the light cone for a Lorentz metric $g$, then $\kappa$ is proportional to the Hodge star operator of $g$. That is, under additional assumptions, the Fresnel surface of $\kappa$ determines the conformal class of $\kappa$. The purpose of this paper is twofold. First, we provide a new proof of this result using Gr\"obner bases. Second, we describe a number of cases where the Fresnel surface does not determine the conformal class of the original $2\choose 2$-tensor $\kappa$. For example, if $\kappa$ is invertible we show that $\kappa$ and $\kappa^{-1}$ have the same Fresnel surfaces.Comment: 23 pages, 1 figur

Management of the Energy Business in the Countries with Developing Economies in the Conditions of the Integration Processes

The present article describes the problems, which are connected with consequences of the ever-increasing influence of the world integration processes upon the national economies in the sphere of energy business. At the global level, this influence is evident as the dependency of governmental budgets of the oil and gas sector from commercial interests of the international corporations, which are owners of the production enterprises in the countries with developing economies. Moreover, it creates new vectors and trends of development both for entire industry and for power markets particularly. Goal of this article is to test this hypothesis on the basis of the corporate structured approach with the help of statistical and financial methods and instruments. Another goal of this article is to determine degree of influence of corporations upon national economies of developing countries in the power-generating sector. As the result, we have succeeded in improvement of the LRD (Limited Risk Distributor), which is the model of the corporate business management. This model is topical one and it is applicable for any energy companies within corporate entities (holdings and groups of companies) in the countries with developing economies. Keywords: energy business; emerging markets; corporations; LRD management business model. JEL Classifications: F23; F36; G34; O13 DOI: https://doi.org/10.32479/ijeep.778

Volume formula for a Z2\mathbb{Z}_2-symmetric spherical tetrahedron through its edge lengths

The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle $\pi$ in the middle points of a certain pair of its skew edges.Comment: 27 pages, 2 figures; enhanced and improved exposition, typos corrected; Arkiv foer Matematik, 201