Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

The interaction of two charges moving in ℝ³ in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either “bounce-back”, “pass-through”, or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into “free” and “trapped” parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.      

Dérivé convexe d’une loi de probabilité et lois stables. Application a un problème d’isomorphisme

The “convex derived set” of a symmetric probability lawF on the real line is defined as the set of limits of laws ∗ _j−1/k ⁿ F(t _j ⁿ η), inf 1≤j≤k _n t _j ⁿ →∞ ifn→∞ and the stable laws it contains are exhibited. A new criterion of stochastic compacity of the set of the powers of a probability law is established. Finally, an isomorphism theorem between somel ^p andL ⁰ spaces is given.

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Laboratoire associé au C.N.R.S. n^o 224 “Processus stochastiques et applications”.     

A consistent value for games with n players and r alternatives

In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a set of axioms (including consistency) which characterizes this value.  The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.  These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning of the players but also on the action chosen by each subset of the partition. Received: April 1994/final version: June 1999     

Solvability of second-order left-invariant differential operators on the Heisenberg group satisfying a cone condition

We discuss local solvability of operators of the form where theV _j are left-invariant vector fields on the Heisenberg group such that [V _j ,V _j+n ]=U for 1≤j≤n andA=(a _jk )=A ₁+iA ₂ is a complex symmetric matrix satisfying the “cone condition” |A ₂|≤CA ₁. The authors acknowledge the support for this work by the European Commission through the European HCM-program “Fourier Analysis” and the TMR network “Harmonic Analysis”.     

Disagreement point axioms and the egalitarian bargaining solution

We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution. Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it to this axiom.     

Darboux-integrable discrete systems

We extend Laplace’s cascade method to systems of discrete “hyperbolic” equations of the form u_i+1,j+1 = f(u_i+1,j, u_i,j+1 , u_i,j), where u_ij is a member of a sequence of unknown vectors, i, j ∊ ℤ. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the well-definedness of the generalized invariant and on its use in the search for solutions and integrals of the system. We give examples of discrete Liouville-type systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 207–219, August, 2008.     

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

Potential Theoretic Approach to Rendezvous Numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x ₁, …, x _n } ⊂ X, there exists some point x ∈ X with the average of the distances d(x,x _j ) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named “the magic number” of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.     

Numerical simulation of sheath plasma by the particle method taking account of coulomb collisions

We study the problem of the behavior of a plasma bounded longitudinally by an absorbing sheath. This model contains charged particles (electrons and ions) moving subject to a self-consistent electrostatic field. New particle pairs are generated in the region of a distributed source. As a numerical model we used the electrostatic “particle-in-cell” method supplemented by the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using the Monte Carlo method. We give a mathematical statement of the problem. The computations were carried out using the direct implicit method with the “explicit limit” time step. The results of numerical simulation of this system are given. We consider the formation and evoluiton of potential structures (multiple weak nonmonotonic double layers). Five figures. Bibliography: 35 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 75–89.     

Potential Theoretic Approach to Rendezvous Numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x ₁, …, x _n } ⊂ X, there exists some point x ∈ X with the average of the distances d(x,x _j ) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named “the magic number” of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.     

Signed words and permutations, V; a sextuple distribution

We calculate the distribution of the sextuple statistic over the hyperoctahedral group B _n that involves the flag-excedance and flag-descent numbers “fexc” and “fdes,” the flag-major index “fmaj,” the positive and negative fixed point numbers “ ” and “ ” and the negative letter number “neg.” Several specializations are considered. In particular, the joint distribution for the pair is explicitly derived.      

The Number of Hierarchical Orderings

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy” these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into m ≤ n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case. This revised version was published online in September 2006 with corrections to the Cover Date.     

On steiner systems

Steiner’s “combinatorial problems” have so far been solved only fork=3 [5, 3] and fork=4 [1,2]. In this paper a complete solution of the problem is given for “closed” Steiner systems, i.e. systems havingn=2 ^k−1−1 elements. Use is made of methods developed by Zaremba [7] for abelian groups. This research was supported by the United States Air Force under Grant No. AF-EOAR-6360 and monitored by the European Office of Aerospace Research.     

2-Adic wavelet bases

It is shown that a “p-adic plane wave” f(t + ω ₁ x ₁ + ... + ω _n x _n ), (t, x ₁, ..., x _n ) ∈ ℚ _p ^{n + 1}, where f is a Bruhat-Schwartz complex-valued test function and max_1≤j≤n |ω _j | _p = 1, satisfies, for any f, a certain homogeneous pseudodifferential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed.     

Suresums

Asuresum is a pair (A, n),A ⊂ {1, ...,n−1}, so that wheneverA is 2-colored some monochromatic set sums ton. A “finite basis” for the suresum (A, n) with |A| ≦c is proven to exist. Forc fixed, it is shown that no suresum (A, n) exist ifn is a sufficiently large prime. Generalizations tor-colorations,r>2, are discussed.     

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V_jⁱ(u) of this system of hydrodynamic type, the metrics g ₁ ^ij(u) and g ₂ ^ij(u), the affinor υ_jⁱ(u) = g ₁ ^is(u)g _2,sj(u), and also the affinors (w _1,n)_jⁱ(u) and (w _2,n)_jⁱ(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.     

The Rankine-Hugoniot conditions and balance laws for <Emphasis Type="Italic">δ</Emphasis>-shocks

New definitions of δ-shock-wave-type solutions are introduced for two (one-dimensional) types of hyperbolic systems of conservation laws. The corresponding Rankine-Hugoniot conditions for δ-shocks are derived and their geometrical interpretation is given. Balance laws connected with “area” mass and momentum transportation for δ-shocks are derived. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 213–229, 2006.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.