109 research outputs found
Factorization in generalized power series
No full textThe field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R((G≤0)) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): Sum_n t^(−1/n) + 1. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = (R, +, 0, ≤) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type α is either ω or of the form ω^ω^α and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case G = R. In the final part of the paper we study the irreducibility of series with finite support
Local connectedness and extension of uniformly continuous functions
No full textA metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146–147 (2005) 339–352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space
The Interpretability Logic of Peano Arithmetic
No full textWe extend Solovay's analysis of the provability logic of Peano Arithmetic, to the case of the interpretability logic
O-minimal spectra, infinitesimal subgroups and cohomology
No full textBy recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal expansion of an ordered field has a normal ``infinitesimal subgroup'' such that the quotient , equipped with the ``logic topology'', is a compact (real) Lie group. Our first result is that the functor sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group and the o-minimal spectrum of . We prove that is a topological quotient of . We thus obtain a natural homomorphism from the cohomology of to the (\v{C}ech-)cohomology of . We show that if satisfies a suitable contractibility conjecture then is acyclic in \v{C}ech cohomology and is an isomorphism. Finally we prove the conjecture in some special cases
Sigma_n Interpretations of modal logic
No full textWe consider the predicative modal provability logic of Peano Arithmetic where the interpretation of each atomic modal formulas is required to be belong to the set of arithmetic formulas of complexity Sigma_0^n. We show that for distinct values of n the corresponding modal formulas form a strict hierarchy and each of them is complete for the class Pi^0_2
Periodicità nei giochi combinatori: il caso di un gioco non ottale.
No full textL'obiettivo di questa tesi è stato quello di analizzare un particolare gioco combinatorio propostomi dal Prof. Alessandro Berarducci, al fine di trovare esplicitamente una strategia vincente per uno dei due giocatori.
Si tratta di un gioco di natura topologica non precedentemente studiato, affine come presentazione, ma non come struttura,
al più famoso Sprouts.
Nel primo capitolo vengono presentati i risultati essenziali nella teoria dei giochi combinatori simmetrici, tra cui il teorema di Sprague-Grundy e il teorema di periodicità per i giochi ottali.
Nel secondo capitolo viene invece presentato il gioco insieme alla relativa analisi, per la quale si è rivelato necessario dimostrare un'estensione del teorema di periodicità per i giochi ottali
A Vietoris–Smale mapping theorem for the homotopy of hyperdefinable sets
No full textResults of Smale (Proc Am Math Soc 8(3): 604–604, 1957) and Dugundji (Fundam Math 66:223–235, 1969) allow to compare the homotopy groups of two
topological spaces X and Y whenever a map f : X → Y with strong connectivity conditions on the fibers is given. We can apply similar techniques to compare the homotopy of spaces living in different categories, for instance an abelian variety over an algebraically closed field, and a real torus. More generally, working in o-minimal expansions of fields, we compare the o-minimal homotopy of a definable set X with the homotopy of some of its bounded hyperdefinable quotients X/E. Under suitable assumption, we show the coincidence of the n-th homotopy and the dimension of X and X/E, each computed in its category. .As a special case, given a definably compact group, we obtain a newproof of Pillay’s group conjecture “dim(G) = dimR(G/G00)” largely independent of the group structure of G. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.
Keyword
COHOMOLOGY OF GROUPS IN O-MINIMAL STRUCTURES: ACYCLICITY OF THE INFINITESIMAL SUBGROUP
No full textAbstract. By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an expansion of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology. 1
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