Multiple separatrix crossing in multi-degree-of-freedom Hamiltonian flows

Summary We study separatrix crossing in near-integrablek-degree-of-freedom Hamiltonian flows, 2 <k < , whose unperturbed phase portraits contain separatrices inn degrees of freedom, 1 <n <k. Each of the unperturbed separatrices can be recast as a codimension-one separatrix in the 2k-dimensional phase space, and the collection of these separatrices takes on a variety of geometrical possibilities in the reduced representation of a Poincaré section on the energy surface. In general 0 l n of the separatrices will be available to the Poincaré section, and each separatrix may be completely isolated from all other separatrices or intersect transversely with one or more of the other available separatrices. For completely isolated separatrices, transitions across broken separatrices are described for each separatrix by the single-separatrix crossing theory of Wiggins, as modified by Beigie. For intersecting separatrices, a possible violation of a normal hyperbolicity condition complicates the analysis by preventing the use of a persistence and smoothness theory for compact normally hyperbolic invariant manifolds and their local stable and unstable manifolds. For certain classes of multi-degree-of-freedom flows, however, a local persistence and smoothness result is straightforward, and we study the global implications of such a local result. In particular, we find codimension-one partial barriers and turnstile boundaries associated with each partially destroyed separatrix. From the collection of partial barriers and turnstiles follows a rich phase space partitioning and transport formalism to describe the dynamics amongst the various degrees of freedom. A generalization of Wiggins' higher-dimensional Melnikov theory to codimension-one surfaces in the multi-separatrix case allows one to uncover invariant manifold geometry. In the context of this perturbative analysis and detailed numerical computations, we study invariant manifold geometry, phase space partitioning, and phase space transport, with particular attention payed to the role of a vanishing frequency in the limit approaching the intersection of the partially destroyed separatrices. The class of flows under consideration includes flows of basic physical relevance, such as those describing scattering phenomena. The analysis is illustrated in the context of a detailed study of a 3-degree-of-freedom scattering problem.     

Coherence and Enumeration of Tilings of 3-Zonotopes

A rhombohedral tiling of a d -zonotope Z is said to be coherent if it may be obtained by projecting the ``top faces' of some (d+1) -zonotope onto Z. We classify those 3 -zonotopes with five or fewer distinct zones which have all rhombohedral tilings coherent, and give concise enumeration formulas for the tilings of the zonotopes in each class. This enumeration relies in equal parts on the theory of oriented matroids and the theory of discriminantal arrangements of hyperplanes. Received August 4, 1997, and in revised form September 3, 1997, and January 16, 1998.     

Localisation des résidus de Baum-Bott, courbes généralisées et K-théorie (I: feuilletages dans \Bbb C2 {\Bbb C}^2 )

Let v be a holomorphic vector field in a neighborhood of a point m ₀ in , which is a non dicritical isolated singularity. Let f = 0 be a reduced equation of the maximal separatrix V through m ₀, v _f the vector field , and the union of separatrices and pseudo-separatrices (i.e. the set of points where v and v _f are colinear). Assuming the foliations defined by v and v _f to be distinct, we prove that the Baum-Bott residue BB(c ₁ ² , v) of v at m ₀, as well as the difference PH(v) - μ between the Poincaré-Hopf index and the Milnor number of V at m ₀, are "localised" near the separatrices and pseudo-separatrices. (The particular case of generalized curves has already been studied in details in [CLS] and [Br]). We also interpret in K-theory the difference PH - μ as well as the GSV index of Gomez Mont-Seade-Verjovski, and we give a caracterisation of generalized curves in this framework, which will enable us to extend this concept in higher dimension. Received: August 25, 2000     

Gauß-Manin determinants for rank 1 irregular connections on curves

Let be a smooth open curve over a field , where k is an algebraically closed field of characteristic 0. Let be a (possibly irregular) absolutely integrable connection on a line bundle L. A formula is given for the determinant of de Rham cohomology with its Gau?-Manin connection . The formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters [7], and this cocycle is shown to exist for connections of arbitrary rank. Received: 13 September 1999 / Published online: 17 August 2001     

The splitting of separatrices,the branching of solutions and non-integrability in the problem of the motion of a spherical pendulum with an oscillating suspension point

The effectiveness of the results obtained previously in [Dovbysh SA. Transversal intersection of separatrices and non-existence of an analytical integral in multidimensional systems. In: Ambrosetti A, Dell Antonio GF, editors. Variational and Local Methods in the Study of Hamiltonian Systems. Singapore, etc: World Scientific; 1995. p. 156–65; Dovbysh SA. Transversal intersection of separatrices, the structure of a set of quasi-random motions and the non-existence of an analytic integral in multidimensional systems. Uspekhi Mat Nauk 1996; 51(4): 153–54; Dovbysh SA. Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points. Collect Math 1999; 50(2): 119–97; Dovbysh SA. Branching of the solutions in the complex domain from the point of view of symbolic dynamics and the non-integrability of multidimensional systems. Dokl Ross Akad Nauk 1998; 361(3): 303–6] on the non-integrability of multidimensional systems is illustrated using the example of the problem of the motion of a spherical pendulum with a suspension point performing small periodic oscillations. With this aim, the splitting of the separatrices of the unstable equilibrium position and the branching of the solutions are investigated. It is shown that the separatrices are split for any law of motion of the suspension point, and a simple criterion of the presence of their transversal intersection is obtained. The validity of the non-integrability result, based on a combination of the conditions related to the splitting of multidimensional separatrices and to the branching of the solutions, is also pointed out.     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

The size of the shadow boundary projection

Among all embedded closed manifolds with positive exterior curvature ≤k the ratio between the (d-1)-Hausdorff measure of the shadow boundary projection and the volume of M ^d is maximized by the sphere of radius 1/k. Received: 22 August 1997 / Revised version: 2 December 1997     

Separatrix splitting from the point of view of symplectic geometry

Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 890–906, June, 1997. Translated by O. V. Sipacheva     

On Random Sections of the Cube

Let f(j,k,n) denote the expected number of j -faces of a random k -section of the n -cube. A formula for f(0,k,n) is presented, and, for j\geq 1 , a lower bound for f(j,k,n) is derived, which implies a precise asymptotic formula for f(n-m,n-l,n) when 1≤ l<m are fixed integers and n→∈fty . Received August 1, 1998, and in revised form December 15, 1998.     

Variational Principles and Sobolev-Type Estimates for Generalized Interpolation on a Riemannian Manifold

The purpose of this paper is to study certain variational principles and Sobolev-type estimates for the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation on a closed (i.e., no boundary), compact, connected, orientable, m -dimensional C ^∞ Riemannian manifold , with C ^∞ metric g _ij . The rate of approximation can be more fully analyzed with rates of approximation given in terms of Sobolev norms. Estimates on the rate of convergence for generalized Hermite and other distributional interpolants can be obtained in certain circumstances and, finally, the constants appearing in the approximation order inequalities are explicit. Our focus in this paper will be on approximation rates in the cases of the circle, other tori, and the 2 -sphere. April 10, 1996. Dates revised: March 26, 1997; August 26, 1997. Date accepted: September 12, 1997. Communicated by Ronald A. DeVore.     

Free monadic Tarski algebras

The free monadic Tarski algebra FMT(n) with a finite set G of n free generators, was determined by A. Figallo in [7]. In this paper we indicate a formula, as a function of n, which is an easier way of calculating the number of elements of FMT(n). Received November 3, 1995; accepted in final form August 26, 1996.     

Convergence of nonstationary cascade algorithms

Summary. A nonstationary multiresolution of is generated by a sequence of scaling functions We consider that is the solution of the nonstationary refinement equations where is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in of the corresponding nonstationary cascade algorithm as k or n tends to It is assumed that there is a stationary refinement equation at with filter sequence h and that The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h. Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999     

The Krein Formula for Generalized Resolvents in Degenerated Inner Product Spaces

 Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data. Received 3 February 1997; in revised form 9 June 1997     

The Linear Arboricity of Series-Parallel Graphs

 The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. A graph is called series-parallel if it contains no subgraphs homeomorphic to K ₄. In this paper, we prove that for any series-parallel graph G having Δ (G)≥3. Since an outerplanar graph is a series-parallel graph, this is also true for any outerplanar graph. Received: August 20, 1997 Revised: March 12, 1999     

A character formula for a family of simple modular representations of $ GL_n $

Let K be an algebraically closed field of finite characteristic p, and let be an integer. In the paper, we give a character formula for all simple rational representations of with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a dominant weight λ is special if there are integers such that and . Indeed, we compute the character of any simple module whose highest weight λ can be written as with all are special. By stabilization, we get a character formula for a family of irreducible rational -modules. Received: June 30, 1997.     

Regularity of Multivariate Refinable Functions

The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A , although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on the spectral properties of A . September 30, 1996. Dates revised: December 1, 1996; February 14, 1997; August 1, 1997; November 11, 1997. Date accepted: November 14, 1997.     

Conformal metrics with prescribed nonpositive Gaussian curvature on {\mathbb R}^2

In this paper, we consider the equation where is a nonpositive function in . A solution u is said to be complete if the conformal metric is complete in . Let Assuming only that , we prove that equation (0.1) possesses infinitely many complete solutions. If in addition, K is assumed to satisfy for some positive constant m, then is also necessary for equation (0.1) to have a complete solution with finite total curvature. We are also able to classify the solution set of equation (0.1) for a wider class of the curvature function K than those considered in [5, 6]. Received October 1, 1997 / Revised version August 10, 1999 / Published online April 6, 2000     

Gradient flows with wildly embedded closures of separatrices

We show that for any n ≥ 4 there exists an n-dimensional closed manifold M ⁿ on which one can define a Morse-Smale gradient flow f ^t with two nodes and two saddles such that the closure of the separatrix of some saddle of f ^t is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.     

Zur Interpretation der Gaußschen Osterformel und ihrer Ausnahmeregeln

In this paper, I present a revised version of Gauss's Easter formula, which is clearer than the original Easter formula and in which certain exceptions are eliminated. I also describe a method for proving calendar algorithms.Copyright 1997 Academic Press.Die Gaußsche Osterformel wird von einem internen Fehler befreit. Dadurch können die viel kritisierten Ausnahmeregeln entfallen. Es wird eine umgebaute Osterformel angegeben, die besser lesbar und verstehbar ist als die ursprüngliche Gaußsche Osterformel. Es wird eine Beweismethode für Kalenderalgorithmen mitgeteilt.Copyright 1997 Academic Press.È stata rielaborata la formula di Gauss per il calcolo della data pasquale così da rendere superflue le eccezioni. È stato proposto un nuovo concetto di tale formula che risulta meglio leggibile ed intellegibile dell'originale formula di Gauss. È stato descritto un metodo per provare gli algoritmi del calendario.