1,062 research outputs found
HR: A System for Machine Discovery in Finite Algebras
We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory
Mathematical applications of inductive logic programming
Accepted versio
On the Notion of Interestingness in Automated Mathematical Discovery
Deciding whether something is interesting or not is of central importance in automated mathematical discovery, as it helps determine both the search space and search strategy for finding and evaluating concepts and conjectures
An Introduction to Mechanized Reasoning
Mechanized reasoning uses computers to verify proofs and to help discover new
theorems. Computer scientists have applied mechanized reasoning to economic
problems but -- to date -- this work has not yet been properly presented in
economics journals. We introduce mechanized reasoning to economists in three
ways. First, we introduce mechanized reasoning in general, describing both the
techniques and their successful applications. Second, we explain how mechanized
reasoning has been applied to economic problems, concentrating on the two
domains that have attracted the most attention: social choice theory and
auction theory. Finally, we present a detailed example of mechanized reasoning
in practice by means of a proof of Vickrey's familiar theorem on second-price
auctions
Corecursive Algebras, Corecursive Monads and Bloom Monads
An algebra is called corecursive if from every coalgebra a unique
coalgebra-to-algebra homomorphism exists into it. We prove that free
corecursive algebras are obtained as coproducts of the terminal coalgebra
(considered as an algebra) and free algebras. The monad of free corecursive
algebras is proved to be the free corecursive monad, where the concept of
corecursive monad is a generalization of Elgot's iterative monads, analogous to
corecursive algebras generalizing completely iterative algebras. We also
characterize the Eilenberg-Moore algebras for the free corecursive monad and
call them Bloom algebras
Combinatorial Hopf algebras in quantum field theory I
This manuscript stands at the interface between combinatorial Hopf algebra
theory and renormalization theory. Its plan is as follows: Section 1 is the
introduction, and contains as well an elementary invitation to the subject. The
rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf
algebra theory and examples, in ascending level of complexity. Part II turns
around the all-important Faa di Bruno Hopf algebra. Section 7 contains a first,
direct approach to it. Section 8 gives applications of the Faa di Bruno algebra
to quantum field theory and Lagrange reversion. Section 9 rederives the related
Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf
algebras of Feynman graphs and, more generally, to incidence bialgebras. In
Section10 we describe the first. Then in Section11 we give a simple derivation
of (the properly combinatorial part of) Zimmermann's cancellation-free method,
in its original diagrammatic form. In Section 12 general incidence algebras are
introduced, and the Faa di Bruno bialgebras are described as incidence
bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us
to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next,
the general algebraic-combinatorial proof of the cancellation-free formula for
antipodes is ascertained; this is the heart of the paper. The structure results
for commutative Hopf algebras are found in Sections 14 and 15. An outlook
section very briefly reviews the coalgebraic aspects of quantization and the
Rota-Baxter map in renormalization.Comment: 94 pages, LaTeX figures, precisions made, typos corrected, more
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