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We obtain normal forms for symmetric and for reversible polynomial automorphisms (polynomial maps that have polynomial inverses) of the plane. Our normal forms are based on the generalized \Henon normal form of Friedland and Milnor. We…

Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special…

Symmetry transformations induce invariances which are frequently described with deep latent variable models. In many complex domains, such as the chemical space, invariances can be observed, yet the corresponding symmetry transformation…

A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random…

The invertibility of integral linear operators is a major problem of both theoretical and practical importance. In this paper we investigate the relation between an operator invertibility and the rank of its integral kernel to develop a…

Fine-tuning pre-trained transformers is a powerful technique for enhancing the performance of base models on specific tasks. From early applications in models like BERT to fine-tuning Large Language Models (LLMs), this approach has been…

In this article we consider the inversion problem for polynomially computable discrete functions. These functions describe behavior of many discrete systems and are used in model checking, hardware verification, cryptanalysis, computer…

A map is given showing that convolutions of independent random variables over a finite group and matrix multiplications of doubly stochastic matrices are homomorphic. As an application, a short proof is given to the theorem that the…

Cycle-consistent training is widely used for jointly learning a forward and inverse mapping between two domains of interest without the cumbersome requirement of collecting matched pairs within each domain. In this regard, the implicit…

In this article, we report, for the first time, broadband multifunctional metasurfaces with more than four distinct functionalities. The constituent meta-atoms combine two different phase change materials, $\mathrm{VO_2}$ and…

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular these functions include all the homogeneous polynomials…

We study the performance of transformers as a function of the number of repetitions of training examples with algorithmically generated datasets. On three problems of mathematics: the greatest common divisor, modular multiplication, and…

Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation…

On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with…

We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has…

Transformers have proven highly effective across various applications, especially in handling sequential data such as natural languages and time series. However, transformer models often lack clear interpretability, and the success of…

We consider a particular class of n-dimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us…

Rotors are common in nature - from rotating membrane-proteins to superfluid-vortices. Yet, little is known about the collective dynamics of heterogeneous populations of rotors. Here, we show experimentally, numerically, and analytically…

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most…

Language models are increasingly capable, yet still fail at a seemingly simple task of multi-digit multiplication. In this work, we study why, by reverse-engineering a model that successfully learns multiplication via \emph{implicit…