Related papers: On integrable generalizations of the pentagram map

The aim of this paper is to study the dimensions and standard part maps between the field of $p$-adic numbers ${{\mathbb Q}_p}$ and its elementary extension $K$ in the language of rings $L_r$. We show that for any $K$-definable set…

We survey results on the problem of covering the space ${\mathbb R}^n$, or a convex body in it, by translates of a convex body. Our main goal is to present a diverse set of methods. A theorem of Rogers is a central result, according to…

An $n \times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection $\mathcal{S}$…

We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…

We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another indeterminate. Furthermore, we classify all such maps for which additionally $JH \cdot H = {\rm…

We prove that the super-linearizability of polynomial systems is preserved by all currently known classes of polynomial automorphisms of $\R^n$. We then establish connections between such automorphisms and a sufficient condition for…

We consider non-singular and Jacobian maps whose components are polynomial in the variable y. We prove that if a map has y-degree one, then it is the composition of a triangular map and a quasi-triangular map. We also prove that…

Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times\mathbb R^k$ if and only if its double point locus $\{(x,y)\in…

We establish the optimal regularity for the distortion of inverses of mappings of finite distortion with logarithm-iterated style subexponentially integrable distortion, which generalizes the Theorem 1. of [J. Gill, Ann. Acad. Sci. Fenn.…

This paper investigates scalable frame in ${\mathbb R}^n$. We define the reduced diagram matrix of a frame and use it to classify scalability of the frame under some conditions. We give a new approach to the scaling problem by breaking the…

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of…

Let $K$ be an algebrically closed field and let $n\geq 1$. If $P\in K[X]=K[X_1,\ldots,X_n]$, $P\neq 0$, we denote by $I(P)$ the support of $P$, which is the finite subset of $\mathbb N^n$ such that $P=\sum_{i\in I(P)}a_iX^i$ with $a_i\in…

We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable…

We introduce a family of generalizations of the pentagram maps related to $Q$-nets. A specific example is considered, and we find the map can be treated as a refactorization mapping in the Poisson-Lie group of pseudo-difference operators.…

We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all assumptions are indispensable. The result…

We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality…

We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. We also define similar variants of this map, that regards alternative models for the modified Macdonald…

In this paper we prove a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds.

Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold…

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful…