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18
CCC forcings and splitting reals
 Israel Journal of Mathematics
"... Abstract. Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i. ..."
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Abstract. Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the Proper Forcing Axiom and is proved using the Pideal dichotomy first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. In the process, I show that under the Pideal dichotomy every weakly distributive ccc complete Boolean algebra carries an exhaustive submeasure, a result which has some interest in its own right. Using a previous theorem of Shelah [Sh1] it follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every non atomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. 1.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
Kingman, category and combinatorics
, 2009
"... Kingman’s Theorem on skeleton limits –passing from limits as n! 1 along nh (n 2 N) for enough h> 0 to limits as t! 1 for t 2 R –is generalized to a Baire/measurable setting via a topological approach. Its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor and another ..."
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Kingman’s Theorem on skeleton limits –passing from limits as n! 1 along nh (n 2 N) for enough h> 0 to limits as t! 1 for t 2 R –is generalized to a Baire/measurable setting via a topological approach. Its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor and another due to Bergelson, Hindman and Weiss is established. As applications, a theory of ‘rational’ skeletons akin to Kingman’s integer skeletons, and more appropriate to a measurable setting, is developed, and two combinatorial results in the spirit of van der Waerden’s celebrated theorem on arithmetic progressions are offered.
On generalized Erdős spaces
, 2008
"... During the last years both Erdős space and complete Erdős space were topologically characterized by Dijkstra and van Mill. Applications include results about Erdős type spaces in pspaces as well as results about Polishable ideals on ω. We present an unifying theorem in terms of sets with a refle ..."
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During the last years both Erdős space and complete Erdős space were topologically characterized by Dijkstra and van Mill. Applications include results about Erdős type spaces in pspaces as well as results about Polishable ideals on ω. We present an unifying theorem in terms of sets with a reflexive relation that among other things contains these apparently dissimilar results as special cases.
AVOIDING FAMILIES AND TUKEY FUNCTIONS ON THE NOWHERE DENSE IDEAL
"... We investigate Tukey functions from the ideal of all closed nowhere dense subsets of 2 N. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove nonexistence of various special types ..."
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We investigate Tukey functions from the ideal of all closed nowhere dense subsets of 2 N. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove nonexistence of various special types of Tukey reductions from the nowhere dense ideal to analytic Pideals. In connection with these results, we study families F of clopen subsets of 2 N with the property that for each nowhere dense subset of 2 N there is a set in F not intersecting it. We call such families avoiding.
THE COSET EQUIVALENCE RELATION AND TOPOLOGIES ON SUBGROUPS
"... The paper studies the structure of the homogeneous space G/H, for G a Polish group and H < G a Borel, not necessarily closed subgroup of G, from the point of view of the theory of definable equivalence relations. It makes a connection between the complexity of the natural coset equivalence relat ..."
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The paper studies the structure of the homogeneous space G/H, for G a Polish group and H < G a Borel, not necessarily closed subgroup of G, from the point of view of the theory of definable equivalence relations. It makes a connection between the complexity of the natural coset equivalence relation associated with G/H and polishability of H, that is, the possibility of introducing a Polish group topology on H respecting its Borel structure. In particular, it is proved that if H is an Abelian Borel subgroup of a Polish group G, then either H is polishable or E1 continuously embeds into the coset equivalence relation induced by H on G. The same conclusion is shown to hold if H is an increasing union of a sequence of polishable subgroups of G.
CARDINAL COEFFICIENTS ASSOCIATED TO CERTAIN ORDERS ON IDEALS
"... Abstract. We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of FréchetUrysohn property and, in a special case, using sequential properties of the sp ..."
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Abstract. We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of FréchetUrysohn property and, in a special case, using sequential properties of the space of finitelysupported probability measures with the weak topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to certain ordering on ideals. 1.
Previous Up Next Article Citations From References: 1 From Reviews: 1
"... Dynamics of a classical Hall system driven by a timedependent AharonovBohm flux. (English summary) J. Math. Phys. 48 (2007), no. 5, 052901, 14 pp. Summary: “We study the dynamics of a classical particle moving in a punctured plane under the influence of a homogeneous magnetic field, an electric ba ..."
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Dynamics of a classical Hall system driven by a timedependent AharonovBohm flux. (English summary) J. Math. Phys. 48 (2007), no. 5, 052901, 14 pp. Summary: “We study the dynamics of a classical particle moving in a punctured plane under the influence of a homogeneous magnetic field, an electric background, and driven by a timedependent singular flux tube through the hole. We exhibit a striking (de)localization effect: when the electric background is absent we prove that a linearly timedependent flux tube opposite to the homogeneous flux eventually leads to the usual classical Hall motion. The particle follows a cycloid whose center is drifting orthogonal to the electric field; if the fluxes are additive, the drifting center eventually gets pinned by the flux tube whereas the kinetic energy grows with the additional flux.” References 1. Asch, J., Benguria, R. D., and ˇSt’ovíček, P., ”Asymptotic properties of the differential equation h 3 (h′ ′ + h′) = 1, ” Asymptotic Anal. 41, 23–40 (2005). MR2124892 (2005j:34069) 2. Asch, J., Hradeck´y, I., and ˇSt’ovíček, P., ”Propagators weakly associated to a family of Hamiltonians and the adiabatic theorem for the Landau Hamiltonian with a timedependent AharonovBohm flux, ” J. Math. Phys. 46, 053303 (2005). MR2143006 (2006a:81033) 3. Avron, J. E., Seiler, R., and Simon, B., ”Charge deficiency, charge transport and comparison of