Related papers: The Search for the Primitive
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…
Let $r \geq 2$ be an integer, $q$ a prime power and $\mathbb{F}_{q}$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A} \subseteq \mathbb{F}_{q^r}$. We prove a sieve…
The theory of evolutionary computation for discrete search spaces has made significant progress in the last ten years. This survey summarizes some of the most important recent results in this research area. It discusses fine-grained models…
The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…
This work considers a natural generalization of primitivity with respect to a language. Given a language $L$, a nonempty word $w$ is said to be $L$-primitive if $w$ is not a proper power of any word in $L$. After ascertaining the number of…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
At any moment in time, evolution is faced with a formidable challenge: refining the already highly optimised design of biological species, a feat accomplished through all preceding generations. In such a scenario, the impact of random…
We extend the relativistic plane--wave impulse approximation formalism to incorporate a specific class of relativistic interference effects for use in describing inclusive electrodisintegration of $^2$H. The role of these ``exchange'' terms…
Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated.
The first-order model theory of modules has been studied for decades. More recently, the model theoretic study of nonelementary classes of modules--especially Abstract Elementary Classes of modules--has produced interesting results. This…
The development of inventions is theorized as a process of searching and recombining existing knowledge components. Previous studies under this theory have examined myriad characteristics of recombined knowledge and their performance…
Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on…
To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary…
Many fundamental questions concerning the emergence and subsequent evolution of eukaryotic exon-intron organization are still unsettled. Genome-scale comparative studies, which can shed light on crucial aspects of eukaryotic evolution,…
We present the ideas behind an algorithm to compute normalizers of primitive groups with non-regular socle in polynomial time. We highlight a concept we developed called permutation morphisms and present timings for a partial implementation…
We discuss various recent advances on weak forms of the Twin Prime Conjecture.
Remarks on mathematical proof and the practice of mathematics.
This study offers a new perspective on the depth-versus-breadth debate in innovation strategy, by modeling inventive search within dynamic collective knowledge systems, and underscoring the importance of timing for technological impact.…
This paper presents new properties of Primitive Pythagorean Triples (PPT) that have relevance in applications where events of different probability need to be generated and in cryptography.