Related papers: Canonical heights and entropy in arithmetic dynami…

We discuss generalizations of the well known concept of canonical transformations for symplectic structures to the case of hyperkahler structures. Different characterizations, which are equivalent in the symplectic case, give raise to…

We prove a lower bound on the relative entropy between two finite-dimensional states in terms of their entropy difference and the dimension of the underlying space. The inequality is tight in the sense that equality can be attained for any…

We introduce notions of vector field and its (discrete time) flow on a chain complex. The resulting dynamical systems theory provides a set of tools with a broad range of applicability that allow, among others, to replace in a canonical way…

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we…

For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined,…

We give an overview of some landmark theorems and recent conjectures in Diophantine Geometry. In the elliptic case, we prove some new bounds for torsion anomalous points and we clarify the implications of several height bounds on the…

We introduce a new canonical height function for Jordan blocks of small eigenvalues for endomorphisms on smooth projective varieties over a number field. We prove that under an assumption on the eigenvalues of the endomorphism on the group…

We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and…

In recent years, the question of whether the ranks of elliptic curves defined over $\mathbb{Q}$ are unbounded has garnered much attention. One can create refined versions of this question by restricting one's attention to elliptic curves…

Many theories of physical interest, which admit a Hamiltonian description, exhibit symmetries under a particular class of non - strictly canonical transformation, known as dynamical similarities. The presence of such symmetries allows a…

We investigate the classical limit of non-Hermitian quantum dynamics arising from a coherent state approximation, and show that the resulting classical phase space dynamics can be described by generalised "canonical" equations of motion,…

A central problem in arithmetic geometry is to construct non-torsion rational points on elliptic curves. We study a canonical quadratic point $\xi_C \in {\rm Jac}(C)$ attached to a smooth non-hyperelliptic curve of genus 4 and use it to…

We present a formalism for which a dissipative system is given by a variational principle. The formalism applies to dynamical systems where its trajectory is monotonic. Subsequently, we derive its Lagrangian and Hamiltonian. From the…

Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued…

We count by height the number of elliptic curves over the rationals that possess an isogeny of degree three.

The aim of this work is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings…

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit…

As previously demonstrated, the entropy production -- a key quantity characterizing the irreversibility of thermodynamic processes -- is related to generation of correlations between degrees of freedom of the system and its thermal…

Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a…

The role of the algebraic method has long been understood in shedding light on the topological structure of sets. However, when the set is a simplicial complex and host to a dynamical process, in particular the trajectory of a canonically…