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The article presents several methods for the arithmetic of finite abelian groups. We introduce a tool - already used by Delsarte in [1] as I found out later - analogous to Dirichlet's convolution to obtain combinatorial results on these…
A companion result of the the Tits alternative for $Out(F_n)$ is proved: Every solvable subgroup of $Out(F_n)$ is finitely generated and virtually abelian.
In this paper, we show how solutions to explicit algebraic systems lead to solutions to infinite families of modular differential equations.
The general entire solution to a linear system of moment differential equations is obtained in terms of a moment kernel function for generalized summability, and the Jordan decomposition of the matrix defining the problem. The growth at…
Put $R=\F[[t_1, \ldots, t_d]])$. We estimate the number of normal subgroups of $\mathrm{SL}_2^1(\F[[t_1, \ldots, t_d]])$ for $p>2$, the number of ideals in the Lie algebra $\Lie(R)$, and the number of ideals in the associative algebra $R$.
We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…
Algebraic data types (ADTs) are a construct classically found in functional programming languages that capture data structures like enumerated types, lists, and trees. In recent years, interest in ADTs has increased. For example, popular…
We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and…
We introduce Riesz Logic, whose models are abelian lattice ordered groups, which generalise Riesz spaces (vector lattices), and show soundness and completeness. Our motivation is to provide a logic for distributional semantics of natural…
(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal…
Natural language understanding is one of the most challenging topics in artificial intelligence. Deep neural network methods, particularly large language module (LLM) methods such as ChatGPT and GPT-3, have powerful flexibility to adopt…
In this short survey we describe recent advances on word equations with non-rational constraints in groups and monoids, highlighting the important role that formal languages play in this area.
The main aim of this paper is to study the growth of solutions of higher order linear differential equations using the concepts of $(\alpha ,\beta ,\gamma )$-order and $(\alpha ,\beta ,\gamma )$-type. We obtain some results which improve…
We show that for any set A in a finite Abelian group G that has at least c |A|^3 solutions to a_1 + a_2 = a_3 + a_4, where a_i belong A there exist sets A' in A and L in G, |L| \ll c^{-1} log |A| such that A' is contained in Span of L and…
This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results…
Large Language Models (LLMs) have been shown to achieve breakthrough performance on complex logical reasoning tasks. Nevertheless, most existing research focuses on employing formal language to guide LLMs to derive reliable reasoning paths,…
We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality…
Pre-trained large language models (LLMs) exhibit powerful capabilities for generating natural text. Evolutionary algorithms (EAs) can discover diverse solutions to complex real-world problems. Motivated by the common collective and…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on…