The hamiltonian property of the consecutive-3 digraph

A consecutive-ifd digraph is a digraph G(d, n, q, r) whose n nodes are labeled by the residues modulo n and a link from node i to node j exists if and only if j  qi + k (mod n) for some k with r ≤ k ≤ r + d − 1. Consecutive-d digraphs are used as models for many computer networks and multiprocessor systems, in which the existence of a Hamiltonian circuit is important. Conditions for a consecutive-d graph to have a Hamiltonian circuit were known except for gcd(n, d) = 1 and d = 3 or 4. It was conjectured by Du, Hsu, and Hwang that a consecutive-3 digraph is Hamiltonian. This paper produces several infinite classes of consecutive-3 digraphs which are not (respectively, are) Hamiltonian, thus suggesting that the conjecture needs modification.     

Algebraic integrability: The Adler-Van Moerbeke approach

In this paper, I present an overview of the active area of algebraic completely integrable systems in the sense of Adler and van Moerbeke. These are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). We make, via the Kowalewski-Painlevé analysis, a study of the level manifolds of the systems. These manifolds are described explicitly as being affine part of abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-Van Moerbeke method’s which will be used is devoted to illustrate how to decide about the algebraic completely integrable Hamiltonian systems and it is primarily analytical but heavily inspired by algebraic geometrical methods. I will discuss some interesting and well known examples of algebraic completely integrable systems: a five-dimensional system, the Hénon-Heiles system, the Kowalewski rigid body motion and the geodesic flow on the group SO(n) for a left invariant metric.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p ² + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrödinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $ \ddot q $ = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $ \ddot q $ = ??(q). The cofactor pair Newton equations admit a Hamilton-Poisson structure in an extended 2n + 1 dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.     

ℓ-distant Hamiltonian walks in Cartesian product graphs

We introduce and study a generalisation of Hamiltonian cycles: an ℓ-distant Hamiltonian walk in a graph G of order n is a cyclic ordering of its vertices in which consecutive vertices are at distance ℓ. Conditions for a Cartesian product graph to possess such an ℓ-distant Hamiltonian walk are given and more specific results are presented concerning toroidal grids.     

Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems

We consider an integrable Hamiltonian system with n degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a G-equivariant way, to the linearized foliation in a neighborhood of a compact singular nondegenerate orbit. We also show that the nondegeneracy condition is not equivalent to the nonresonance condition for smooth systems.     

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.     

The multiplicative Weyl functional calculus

The Weyl calculus discussed in the author's previous papers starts with a fixed set of n noncommuting self-adjoint operators and associates an operator to a real function of n variables. The calculus is not multiplicative with respect to point-wise multiplication of functions. However, if the n self-adjoint operators generate a unitary Lie group representation, a “skew product” of functions can be defined which yields multiplicativity. This skew product depends only on the Lie group, not on the particular representation. In the case of the Heisenberg group, this skew product makes it possible to write the Schrödinger equation as an integro-differential equation on the phase plane. Strong convergence of the dynamical group, as Planck's constant goes to zero, to the classical Hamiltonian flow is proved under various conditions on the Hamiltonian.     

A-posteriori KAM theory with optimal estimates for partially integrable systems

In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional $n?d$ independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.     

The Hamiltonian property of linear functions

Consider a digraph G(n, q, r) with n nodes and n links i → qi + r, i = 0, 1,…, n − 1, where q and r are given. The topologies of many computer networks use G(n, q, r) as basic building block. A digraph is called Hamiltonian if it contains a circuit spanning all nodes. The Hamiltonian property of a network topology provides the capability of configuring the interconnection network as a linear array, which is the configuration with the broadest practical significance, of either n − 1 or n nodes in the presence of a single faulty node or link. In this paper we give necessary and sufficient conditions for G(n, q, r) to be Hamiltonian.     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.     

On the number of Hamiltonian cycles in tournaments

The main results assert that the minimum number of Hamiltonian bypasses in a strong tournament of order n and the minimum number of Hamiltonian cycles in a 2-connected tournament of order n increase exponentially with n. Furthermore, the number of Hamiltonian cycles in a tournament increases at least exponentially with the minimum outdegree of the tournament. Finally, for each k?1 there are infinitely many tournaments with precisely k Hamiltonian cycles.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

Multidimensional Latin bitrades

A subset of the n-dimensional k-valued hypercube is a unitrade or united bitrade whenever the size of its intersections with the one-dimensional faces of the hypercube takes only the values 0 and 2. A unitrade is bipartite or Hamiltonian whenever the corresponding subgraph of the hypercube is bipartite or Hamiltonian. The pair of parts of a bipartite unitrade is an n-dimensional Latin bitrade. For the n-dimensional ternary hypercube we determine the number of distinct unitrades and obtain an exponential lower bound on the number of inequivalent Latin bitrades. We list all possible n-dimensional Latin bitrades of size less than 2 ⁿ⁺¹. A subset of the n-dimensional k-valued hypercube is a t-fold MDS code whenever the size of its intersection with each one-dimensional face of the hypercube is exactly t. The symmetric difference of two single MDS codes is a bipartite unitrade. Each component of the corresponding Latin bitrade is a switching component of one of these MDS codes. We study the sizes of the components of MDS codes and the possibility of obtaining Latin bitrades of a size given from MDS codes. Furthermore, each MDS code is shown to embed in a Hamiltonian 2-fold MDS code.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

Integrability theorems for a class of integral transforms

It is shown under weak hypotheses that systems of 2n linear differential equations in 2n variables generate sets of identities similar in structure to the classical trigonometric identities. For clarity of exposition only the case n = 1 is actually treated, but all final equations are written in such a manner as to be directly applicable to matrix systems (n > 1). These identities allow one to avoid, in a very simple way, certain difficulties which often occur in the integration of the Riccati equations arising from application of the invariant imbedding method to two point boundary value problems associated with such linear systems. The overall usefulness of the imbedding method is thereby considerably extended. One analytical and one numerical example are given to illustrate the actual use of these identities.     

Hamiltonian decompositions of complete regular s-partite graphs

In this paper we give a procedure by which Hamiltonian decompositions of the s-partite graph K_n,n,…,n, where (s − 1)n is even, can be constructed. For 2t⩽s, 1⩽a₁⩽…⩽a_t⩽n, we find conditions which are necessary and sufficient for a decomposition of the edge-set of K_a1a2…,a_t into (s − 1)n/2 classes, each class consisting of disjoint paths, to be extendible to a Hamiltonian decomposition of the complete s-partite graph K_rmn,n,…,n so that each of the classes forms part of a Hamiltonian cycle.