211 research outputs found
Theta blocks related to root systems
Gritsenko, Skoruppa and Zagier associated to a root system a theta block
, which is a Jacobi form of lattice index. We classify the theta
blocks of -order and show that their Gritsenko lift is a
strongly-reflective Borcherds product of singular weight, which is related to
Conway's group . As a corollary we obtain a proof of the
theta block conjecture by Gritsenko, Poor and Yuen for the pure theta blocks
obtained as specializations of the functions .Comment: Final version, published in Math. Ann.; updated acknowledgemen
Some notes on impartial games and NIM dimension
Tese de doutoramento, Matemática (Análise Numérica e Matemática Computacional), Universidade de Lisboa, Faculdade de Ciências, 2010.Disponível no documento
Closure algorithms and the star-height problem of regular languages
Imperial Users onl
Computing backwards with Game of Life, part 1: wires and circuits
Conway's Game of Life is a two-dimensional cellular automaton. As a dynamical
system, it is well-known to be computationally universal, i.e.\ capable of
simulating an arbitrary Turing machine. We show that in a sense taking a single
backwards step of Game of Life is a computationally universal process, by
constructing patterns whose preimage computation encodes an arbitrary
circuit-satisfaction problem, or (equivalently) any tiling problem. As a
corollary, we obtain for example that the set of orphans is coNP-complete,
exhibit a -periodic configuration that admits a preimage but
no periodic one, and prove that the existence of a preimage for a periodic
point is undecidable. Our constructions were obtained by a combination of
computer searches and manual design.Comment: 28 pages, 10 figures in main text. 11 pages, 20 figures in appendix.
Accompanied by two GitHub repositories containing programs and auxiliary dat
Conway’s Circle Theorem: a short proof, enabling generalization to polygons
John Conway’s Circle Theorem is a gem of plane geometry: the six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs, even adorned Mathcamp T-shirts. We present a short proof that views the extended sides as equal tangents of the incircle, a perspective that enables generalization to polygons.Accepted manuscrip
On Kleene algebras of ternary co-relations
In this paper we investigate identities satisfied by a class of algebras made of ternary co-relations - contravariant ("arrow-reversed") analogues of binary relations. These algebras are equipped with the operations of union, co-relational composition, iteration, converse and the empty co-relation and the so-called diagonal co-relation as constants. Our first result is that the converse-free part of the corresponding equational theory consists precisely of Kleenean equations for relations, or, equivalently, for (regular) languages. However, the rest of the equations, involving the symbol of the converse, are relatively axiomatized by involution axioms only, so that the co-relational converse behaves more like the reversal of languages, rather than the relational converse. Actually, the language reversal is explicitely used to prove this result. Therefore, we conclude that co-relations can offer a better framework than relations for the mathematical modeling of formal languages, as well as many other notions from computer science
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