The optimal upper bound of the number of generalized Euler configurations

Consider a generalized 3-body problem. The attraction force between any two bodies is proportional to the two masses and the b-th power of the mutual distance. Albouy and Fu have obtained the optimal upper bound of the number of generalized Euler configurations for the cases b 1 and b = 2, 3. This paper obtains the optimal upper bound for the remaining real values of b in a systematic way.     

Equilibrium configurations of point vortices on a sphere

Configurations of point vortices on the sphere are considered in which all vortex velocities are zero. A sharp upper bound for the number of equilibria lying on a great circle is found, valid for generic circulations, and some unusual equilibrium configurations with a free real parameter are described. Equilibria of rings (vortices evenly spaced along circles of latitude) are also discussed. All equilibrium configurations of four vortices are determined.      

Relative Equilibria of the (1+N)-Vortex Problem

We examine existence and stability of relative equilibria of the n-vortex problem specialized to the case where N vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued via the Implicit Function Theorem, and its linear stability is determined by the eigenvalues of the Hessian matrix of the potential. For N≥3 there are at least three distinct families of critical points associated to the limiting problem. Assuming nondegeneracy, one of these families continues to a linearly stable class of relative equilibria with small and large circulation of the same sign. This class becomes unstable as the small circulation passes through zero and changes sign. Another family of critical points which is always nondegenerate continues to a configuration with small vortices arranged in an N-gon about the strong central vortex. This class of relative equilibria is linearly unstable regardless of the sign of the small circulation when N≥4. Numerical results suggest that the third family of critical points of the limiting problem also continues to a linearly unstable class of solutions of the full problem independent of the sign of the small circulation. Thus there is evidence that linearly stable relative equilibria exist when the large and small circulation strengths are of the same sign, but that no such solutions exist when they have opposite signs. The results of this paper are in contrast to those of the analogous celestial mechanics problem, for which the N-gon is the only relative equilibrium for N sufficiently large, and is linearly stable if and only if N≥7.     

Relative equilibrium and collapse configurations of four point vortices

Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.      

Symmetric block bases in finite-dimensional normed spaces

It is shown that for 1 ≦p < ∞, any basisC-equivalent to the unit vector basis ofl _p ⁿ has a (1 + ε)-symmetric block basis of cardinality proportional ton/logn. When 1 <p < ∞, an upper bound proportional ton log logn/logn is also obtained. These results extend results of Amir and Milman in [2].     

A recursive algorithm for the infinity-norm fixed point problem

We present the PFix algorithm for the fixed point problem f(x)=x on a nonempty domain [a,b], where d1, , and f is a Lipschitz continuous function with respect to the infinity norm, with constant q1. The computed approximation satisfies the residual criterion , where >0. In general, the algorithm requires no more than ∑_i=1^dsⁱ function component evaluations, where s≡max(1,log₂(||b−a||_∞/))+1. This upper bound has order as →0. For the domain [0,1]^d with <0.5 we prove a stronger result, i.e., an upper bound on the number of function component evaluations is , where r≡log₂(1/). This bound approaches as r→∞ (→0) and as d→∞. We show that when q<1 the algorithm can also compute an approximation satisfying the absolute criterion , where x^* is the unique fixed point of f. The complexity in this case resembles the complexity of the residual criterion problem, but with tolerance (1−q) instead of . We show that when q>1 the absolute criterion problem has infinite worst-case complexity when information consists of function evaluations. Finally, we report several numerical tests in which the actual number of evaluations is usually much smaller than the upper complexity bound.     

Regularity Index of Fat Points in the Projective Plane

In this paper we determine a new upper bound for the regularity index of fat points of P², without requiring any geometric condition on the points. This bound is intermediate between Segre′s bound, that holds for points in the general position, and the more general bound, that is attained when the points are collinear: in fact, both of these bounds can be recovered as particular cases. Furthermore, our bound cannot, in general, be sharpened: in fact, it is attained if there are either many collinear points or collinear points with high multiplicities.     

Linear stability of relative equilibria in the charged three-body problem

A relative equilibrium is a periodic orbit of the n-body problem that rotates uniformly maintaining the same central configuration for all time. In this paper we generalize some results of R. Moeckel and we apply it to study the linear stability of relative equilibria in the charged three-body problem. We find necessary conditions to have relative equilibria linearly stable for the collinear charged three-body problem, for planar relative equilibria we obtain necessary and sufficient conditions for linear stability in terms of the parameters, masses and electrostatic charges. In the last case we obtain a stability inequality which generalizes the Routh condition of celestial mechanics. We also proof the existence of spatial relative equilibria and the existence of planar relative equilibria of any triangular shape.     

Stationary Configurations of Point Vortices on a Cylinder

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ₁ and Γ₂ (Γ₂ = ?μΓ₁) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.     

Saari's conjecture for the collinear -body problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian -body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

     

A minimal area problem in conformal mapping II

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU. For 0<r<1 andr(1+r)⁻²<b<r(1−r)⁻², letS(r, b) be the subclass of functionsf∈S such that |f(r)|=b. In Theorem 1, we solve the problem of minimizing the Dirichlet integral inS(r, b). The first main ingredient of the solution is the establishment of sufficient regularity of the domains onto whichU is mapped by extremal functions, and here techniques of symmetrization and polarization play an essential role. The second main ingredient is the identification of all Jordan domains satisfying a certain kind of functional equation (called “quadrature identities”) which are encountered by applying variational techniques. These turn out to be conformal images ofU by mappings of a special form involving a logarithmic function. In Theorem 2, this aspect of our work is generalized to encompass analogous minimal area problem when a larger number of initial data are prescribed. The third author thanks for its hospitality the Mittag-Leffler Institute of Royal Swedish Academy of Sciences where this work was finalized. This author was supported in part by the Swedish Institute and by the Russian Fund for Fundamental Research, grant no. 97-01-00259.     

Maximum diameter of regular digraphs

We prove that every r-biregular digraph with n vertices has its directed diamter bounded by (3n - r - 3)/(r +1). We show that this bound is tight for directed as well as for undirected graphs. The upper bound remains valid for Eulerian digraphs with minimum outdegree r. © 1929 John Wiley & Sons, Inc.     

Homographic solutions of the n-body problem in E4

We prove the existence of many homographic solutions of the n-body problem in E⁴ by topological methods. Homographic solutions are associated with relative equilibria. Homothetic solutions always give rise to central configurations. In Euclidean space E⁴ central configurations are a proper subset of the relative equilibria for any n ? 3 and for any (m_i)?R₊ⁿ. We compare the existence and classification of homographic solutions of the n-body problem in E³ with the Newtonian potential and that of homographic solutions of the n-body problem in E⁴. Classifying relative equilibria leads to classifying homographic solutions.     

Dynamics of perturbed equilateral and collinear configurations of three point vortices

Using the technique of asymptotic expansions, we calculate trajectories of three point vortices in the vicinity of stable equilateral or collinear configurations. We show that in an appropriate rotating coordinate system each vortex moves in an elliptic orbit. The orbits of the vortices have equal eccentricities. The angle and ratio between the major axes of any two orbits have a simple analytic representation.      

A Reilly inequality for the first Steklov eigenvalue

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p₁ of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.     

On Ramanujan asymptotic expansions and inequalities for hypergeometric functions

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r ²) < (2−r ²)F(a,b;a +b;r ²) holds for all r∈(0,1).     

Collinear Points in Permutations

Consider the following problem: how many collinear triples of points must a transversal of have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n − 1)/4 and (n − 1)/2, and consider an analogous question for collinear quadruples. We conjecture that the upper bound is the truth and suggest several other interesting problems in this area.Received August 29, 2004     

Symmetry of the restricted 4 + 1 body problem with equal masses

We consider the problem of symmetry of the central configurations in the restricted 4 + 1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1–3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4 + 1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function ϕ(s) = −s ^k, with k < 0) which are crucial in the proof of the symmetry.