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## Устойчивые двумерные торы в ловушке Пеннинга при комбинированном частотном резонансе

We study the planar Penning traps in a resonance mode. The axial symmetry of the system is violated by deviation of the magnetic field from the trap axis at a small angle (the small parameter in the given model). The geometry of planar electrodes and their electric potentials are made consistent to reach a combined resonance, in both prime and subprime Hamiltonians under the small parameter expansion. In such a double-resonance regime we make the double averaging and derive the explicit formulas for the dependence of the averaged Hamiltonian on the controlling parameters of the trap. After the double reduction with respect to the primary and the secondary symmetry algebras, for the reduced Hamiltonian, an algorithm and explicit formulas for calculating all equilibrium points, explicit formulas for the energies and for the Hessians at these points in terms of the initial controlling parameters of the trap are obtained. In the original six-dimensional phase space the stable equilibrium points are related to invariant two-dimensional tori winded by near-periodic trajectories of a charge moving in the trap.

We discuss the general opportunity to create (asymptotically) a comletely integrable system from the original perturbed system by inserting additional perturbing terms. After such an artificial insertion, there appears an opportunity to make the secondary averaging and secondary reduction of the original system. Thus, by this way, the $3D$-system becomes $1$-dimensional. We demonstrate this approach by the example of a resonance Penning trap.

In this paper the basic properties of different types of equilibrium concepts in antagonistic games with various preference structures are considered.

This is the first book on the U.S. presidential election system to analyze the basic principles underlying the design of the existing system and those at the heart of competing proposals for improving the system. The book discusses how the use of some election rules embedded in the U.S. Constitution and in the Presidential Succession Act may cause skewed or weird election outcomes and election stalemates. The book argues that the act may not cover some rare though possible situations which the Twentieth Amendment authorizes Congress to address. Also, the book questions the constitutionality of the National Popular Vote Plan to introduce a direct popular presidential election de facto, without amending the Constitution, and addresses the plan’s “Achilles’ Heel.” In particular, the book shows that the plan may violate the Equal Protection Clause from the Fourteenth Amendment of the Constitution. Numerical examples are provided to show that the counterintuitive claims of the NPV originators and proponents that the plan will encourage presidential candidates to “chase” every vote in every state do not have any grounds. Finally, the book proposes a plan for improving the election system by combining at the national level the “one state, one vote” principle – embedded in the Constitution – and the “one person, one vote” principle. Under this plan no state loses its current Electoral College benefits while all the states gain more attention of presidential candidates.

In this paper we consider games with preference relations. The main optimality concept for such games is concept of equilibrium. We introduce a notion of homomorphism for games with preference relations and study a problem concerning connections between equilibrium points of games which are in a homomorphic relation. The main result is finding covariantly and contravariantly complete families of homomorphisms.

The paper proposes two new approaches to designing efficient mathematical tools for quantitatively analyzing decision-making processes that small and medium price-taking traders undergo in forming and managing their portfolios of financial instruments traded in a stock exchange. Two mathematical models underlying these approaches are considered. If the trader can treat price changes for each financial instrument of her interest as those of a random variable with a known (for instance, a uniform) probability distribution, one of these models allows the trader to formulate the problem of finding an optimal composition of her portfolio as an integer programming problem. The other model is suggested to use when the trader does not possess any particular information on the probability distribution of the above-mentioned random variable for financial instruments of her interest while being capable of estimating the areas to which the prices of groups of financial instruments (being components of finite-dimensional vectors for each group) are likely to belong. When each such area is a convex polyhedron described by a finite set of compatible linear equations and inequalities of a balance kind, the use of this model allows one to view the trader’s decision on her portfolio composition as that of a player in an antagonistic game on sets of disjoint player strategies. The payoff function of this game is a sum of a linear and a bilinear function of two vector arguments, and the trader’s guaranteed financial result in playing against the stock exchange equals the exact value of the maximin of this function. This value, along with the vectors at which it is attained, can be found by solving a mixed programming problem. Finding an upper bound for this maximin value (and the vectors at which this upper bound is attained) is reducible to finding saddle points in an auxiliary antagonistic game with the same payoff function on convex polyhedra of disjoint player strategies. These saddle points can be calculated by solving linear programming problems forming a dual pair.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.