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We address the problem of admissibility of pre-Lie structures associated with a given Lie algebra, particularly, semisimple Lie algebras over ${\mathbb C}$. Such structures are collectively referred to as Lie-admissible algebras, which are…
A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA $N$, there exists a process algebraic term $p$ such that its semantics is…
We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a similarity matrix for $n$ data points requires $\Omega(n^2)$ similarity computations. This quadratic…
Large language models (LLMs) have complicated internal dynamics, but induce representations of words and phrases whose geometry we can study. Human language processing is also opaque, but neural response measurements can provide (noisy)…
The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the…
Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that…
Human language has a distinct systematic structure, where utterances break into individually meaningful words which are combined to form phrases. We show that natural-language-like systematicity arises in codes that are constrained by a…
A compatible associative algebra is a vector space equipped with two associative multiplication structures that interact in a certain natural way. This article presents the classification of these algebras with dimension less than four, as…
One of the ultimate goals for linguists is to find universal properties in human languages. Although words are generally considered as representing arbitrary mapping between linguistic forms and meanings, we propose a new universal law that…
Logical relations constitute a key method for reasoning about contextual equivalence of programs in higher-order languages. They are usually developed on a per-case basis, with a new theory required for each variation of the language or of…
This paper investigates whether large language models (LLMs) utilize numerical attributes encoded in a low-dimensional subspace of the embedding space when answering questions involving numeric comparisons, e.g., Was Cristiano born before…
We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…
Large Language Models (LLMs) excel at understanding natural language but struggle with explicit commonsense reasoning. A recent trend of research suggests that the combination of LLM with robust symbolic reasoning systems can overcome this…
We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…
We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a…
There are at least two kinds of similarity. Relational similarity is correspondence between relations, in contrast with attributional similarity, which is correspondence between attributes. When two words have a high degree of attributional…
An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described.
Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism…
Let A be a finite-dimensional associative algebra and $\phi$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps are obtained as special cases of well-known symmetric linear functions on the endomorphism…
Regular languages -- the languages accepted by deterministic finite automata -- are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we…