Related papers: Decoding Rauzy Induction: An Answer to Bufetov's G…

We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the…

Transformer models underpin many recent advances in practical machine learning applications, yet understanding their internal behavior continues to elude researchers. Given the size and complexity of these models, forming a comprehensive…

In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a…

We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are…

We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in an Euclidean space which is a finite topological extension of a translation on a torus.This generalizes the pioneer works of Rauzy and…

An open problem in convex geometry asks whether two simplices $A,B\subseteq\mathbb{R}^d$, both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming $A$ into $B$ while maintaining…

We show that every transformation is disjoint from almost every interval exchange transformation (IET), answering a question of Bufetov. In particular, we prove that almost every pair of IETs is disjoint. It follows that the product of…

We explore the Iterative Inference Hypothesis (IIH) within the context of transformer-based language models, aiming to understand how a model's latent representations are progressively refined and whether observable differences are present…

We present the first explicit example of an interval exchange transformation with flips (FIET) possessing three distinct invariant ergodic measures. The proof is based on a generalization of M. Keane's method, using the Rauzy induction…

The eigendecomposition of a matrix is the central procedure in probabilistic models based on matrix factorization, for instance principal component analysis and topic models. Quantifying the uncertainty of such a decomposition based on a…

In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In…

We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one…

By work of De Concini, Kac and Procesi the irreducible representations of the non-restricted specialization of the quantized enveloping algebra of the Lie algebra g at the roots of unity are parametrized by the conjugacy classes of a group…

We consider continuous, translation-commuting transformations of compact, translation-invariant families of mappingsfrom finitely generated groups into finite alphabets. It is well-known that such transformations and spaces can be described…

This paper deals with Fountain codes, and especially with their encoding matrices, which are required here to be invertible. A result is stated that an encoding matrix induces a permutation. Also, a result is that encoding matrices form a…

We investigate the relation between bifix codes and interval exchange transformations. We prove that the class of natural codings of regular interval echange transformations is closed under maximal bifix decoding.

In this article we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering…

Seymour's famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called $k$-sums starting from network matrices and their transposes and…

The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned…

We are interested in the problem of translating between two representations of closure systems, namely implicational bases and meet-irreducible elements. Albeit its importance, the problem is open. Motivated by this problem, we introduce…