Related papers: Learning Algebraic Structures: Preliminary Investi…
In this paper we consider a problem of searching a space of predictive models for a given training data set. We propose an iterative procedure for deriving a sequence of improving models and a corresponding sequence of sets of non-linear…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with…
We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of…
We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting…
In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct…
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using…
This book has eleven chapters. Chapter one describes all types of natural class of intervals and the arithmetic operations on them. Chapter two introduces the semigroup of natural class of intervals using R or Zn and study the properties…
Classical and exceptional Lie algebras and their representations are among the most important tools in the analysis of symmetry in physical systems. In this letter we show how the computation of tensor products and branching rules of…
We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is…
Humans possess a remarkable capacity to recognize and manipulate abstract structure, which is especially apparent in the domain of geometry. Recent research in cognitive science suggests neural networks do not share this capacity,…
We present here some results of applying the Cayley-Dickson process to certain alternative algebras (notably built upon Galois fields and congruence rings), in a manner which might yield new building blocks for cryptographic systems. We…
We review supervised learning and deep neural network design for learning membership on algebraic varieties. We demonstrate that these trained artificial neural networks can predict the entanglement type for quantum states. We give examples…
Given a large number of unlabeled face images, face grouping aims at clustering the images into individual identities present in the data. This task remains a challenging problem despite the remarkable capability of deep learning approaches…
Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…
This chapter reviews the application of Artificial Intelligence (AI) techniques to the study of galaxy clusters, covering both theoretical developments and their use as tools to infer cluster properties from a variety of observational…
Algebraic geometry for groups and Lie algebraic has been recently defined and studied by many authors on the purpose to study set defined by algebraic equations on abstract groups and Lie algebras. The purpose of this paper is to present a…
We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically…
We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from…