1,721,057 research outputs found
The Euler characteristic and valuations on MV-algebras
No full textEvery finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each a ∈ A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero. © 2014 Versita Warsaw and Springer-Verlag Wien
The Lebesgue state of a unital abelian lattice-ordered group
No full textVarious combinatorial equivalents are given to the Lebesgue state in archimedean lattice-ordered groups with order-unit. The proofs use piecewise linear functions on polyhedra with rational vertices
Optimal comparison strategies in Ulam's searching game with two errors
No full textAbstractSuppose x is an n-bit integer. By a comparison question we mean a question of the form “does x satisfy either condition a ⩽x ⩽b or c ⩽x ⩽d?”. We describe strategies to find x using the smallest possible number q(n) of comparison questions, and allowing up to two of the answers to be erroneous. As proved in this self-contained paper, with the exception of n = 2, q(n) is the smallest number q satisfying Berlekamp's inequality 2q⩾2nq2+ q + 1. This result would disappear if we only allowed questions of the form “does x satisfy the condition a⩽x⩽b?”. Since no strategy can find the unknown x ∈ {0,1,…,2n −1} with less than q(n) questions, our result provides extremely simple optimal searching strategies for Ulam's game with two lies—the game of Twenty Questions where up to two of the answers may be erroneous
Lattice-ordered Abelian groups and Schauder bases of unimodular fans
No full textBaker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support of a fan |S|. A unimodular fan D over |S| determines a Schauder basis of G: its elements are the minimal positive free generators of the pointwise ordered group of D-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |S|: its maximal cones are the domains of linearity of the elements of H. The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, G is finitely generated projective iff it can be presented by a purely lattice-theoretical word
An algorithmic desingularization of 3-dimensional toric varieties
No full textWe present an algorithmic procedure to desingularize every 3-dimensional toric variety, while keeping under control the Euler characteristic of the varieties computed during the process. We prove that our upper bounds for the Euler
characteristic of the desingularized toric varieties are the best possible
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