Related papers: A three-dimensional generalization of QRT maps
To visualize a higher dimensional object it is convenient to consider its two-dimensional cross-sections. The set of quantum states for a three level system has eight dimensions. We supplement a recent paper by Goyal et al by considering…
Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discretization method exactly preserves a nearby modifed integral.…
We constructed involutions for a Halphen pencil of index 2, and proved that the birational mapping corresponding to the autonomous reduction of the elliptic Painlev\'e equation for the same pencil can be obtained as the composition of two…
Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques…
This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the…
We derive three-dimensional integrable mappings which have two invariants.
In this paper first we give a partial answer to a question of L. Moln\'ar and W. Timmermann. Namely, we will describe those linear (not necessarily bijective) transformations on the set of self-adjoint matrices which preserve a unitarily…
We describe pretty examples of derived equivalences and autoequivalences of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds are chosen to be special, so they have an associated K3 surface. Thus a pencil…
We discuss the singularity structure of Kahan discretizations of a class of quadratric vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an…
Three-dimensional (3D) mappings are fundamental in various scientific and engineering applications, including computer-aided engineering (CAE), computer graphics, and medical imaging. They are typically represented and stored as…
We study the congruence of bitangent lines of an irreducible surface in the 3-dimensional projective space in arbitrary characteristic, with special attention to quartic surfaces with rational double points and, in particular, Kummer…
We review combinatorial quantum gravity, an approach which combines Einstein's idea of dynamical geometry with Wheeler's "it from bit" hypothesis in a model of dynamical graphs governed by the coarse Ollivier-Ricci curvature. This drives a…
A novel family of integrable third order maps is presented. Each map possesses, by construction, a pair of rational invariants and a commuting map from the same class. The 3-dimensional invariant curve is parametrized, in general, by an…
It is well known that the dynamical system determined by a Quispel-Roberts-Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on ${\mathbb{CP}}^1 \times {\mathbb{CP}}^1$. In most cases this pencil is elliptic, i.e.…
Notion of an open system of second order is introduced. Characteristic function for such an open system is obtained. Model representations of a quadratic non-self-adjoint operator pencil are found.
In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu--Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close…
We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which…
I construct some smooth Calabi-Yau threefolds in characteristic two and three that do not lift to characteristic zero. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian…
We construct some examples of D=3, N=4 GW theory and N=5 superconformal Chern-Simons matter theory by using the covariantly constant curvature of a quaternionic-Kahler manifold to construct the symplectic 3-algebra in the theories.…
In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…