Related papers: Algebraic curves as a source of separable multi-Ha…
We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of the Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (corner…
In this article we investigate stationary systems of the AKNS hierarchy. We prove that all these systems are classical separable St\"ackel systems. The explicit transformation from jet variables to separation coordinates is obtained by…
This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect…
For a linear non-Hermitian system, I demonstrate that a Hamiltonian can be constructed such that the non-Hermitian equations can be expressed exactly in the form of Hamilton's canonical equations. This is first shown for discrete systems…
We show that, when applied to any non-canonical Hamiltonian system, any integrator that is symplectic for canonical Hamiltonian problems is actually conjugate symplectic for the non-canonical structure. This result is useful because it…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
In this paper, we prove a lemma on logarithmic derivative for holomorphic curves from annuli into K\"{a}hler compact manifold and. As its application, a second main theorem for holomophic curves from annuli into semi abelian varieties…
A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed…
Let $C$ be a complex, reduced, locally planar curve. We extend the results of Rennemo arXiv:1308.4104 to reducible curves by constructing an algebra $A$ acting on $V=\bigoplus_{n\geq 0} H_*(C^{[n]}, \mathbb{Q})$, where $C^{[n]}$ is the…
Properties of the recently reported homogeneous Hilbert curves are deduced and reported. The nature of the affine transformations involved in the construction of the Hilbert curves is explored. The analytical representation of proper and…
Hamiltonian bifurcations in the context of noncanonical Hamiltonian matter models are described. First, a large class of 1 + 1 Hamiltonian multi-fluid models is considered. These models have linear dynamics with discrete spectra, when…
By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of…
It is shown that all 3-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional…
In a previous paper, we developed a table of components of algebraic solutions of a system of equations generated by an inhomogeneous proper-value equation involving K\"ahler's total angular momentum. This table looks as if it were a…
This article is meant as a mathematical appendix or comment on [BT]. We first consider the notion of transcritical bifurcations of fixed points of general area-preserving maps, and then adress some questions related to [BT] on bifurcation…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…
In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry…
The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical…
An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve a quantum computer into an entanglement of states which map onto the set of all possible paths…