402 research outputs found

    Effectiveness of Hindman's theorem for bounded sums

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    We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let HTkn\mathsf{HT}^{\leq n}_k denote the assertion that for each kk-coloring cc of N\mathbb{N} there is an infinite set XNX \subseteq \mathbb{N} such that all sums xFx\sum_{x \in F} x for FXF \subseteq X and 0<Fn0 < |F| \leq n have the same color. We prove that there is a computable 22-coloring cc of N\mathbb{N} such that there is no infinite computable set XX such that all nonempty sums of at most 22 elements of XX have the same color. It follows that HT22\mathsf{HT}^{\leq 2}_2 is not provable in RCA0\mathsf{RCA}_0 and in fact we show that it implies SRT22\mathsf{SRT}^2_2 in RCA0\mathsf{RCA}_0. We also show that there is a computable instance of HT33\mathsf{HT}^{\leq 3}_3 with all solutions computing 00'. The proof of this result shows that HT33\mathsf{HT}^{\leq 3}_3 implies ACA0\mathsf{ACA}_0 in RCA0\mathsf{RCA}_0

    The scarcity of products in βS ∖ S

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    Let S be a discrete semigroup and let the Stone–Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center. Let S ⁎ = β S ∖ S . Algebraically, the set of products S ⁎ S ⁎ tends to be rather large, since it often contains the smallest ideal of βS. We establish here sufficient conditions involving mild cancellation assumptions and assumptions about the cardinality of S for S ⁎ S ⁎ to be topologically small, that is for S ⁎ S ⁎ to be nowhere dense in S ⁎ , or at least for S ⁎ ∖ S ⁎ S ⁎ to be dense in S ⁎ . And we provide examples showing that these conditions cannot be significantly weakened. These extend results previously known for countable semigroups. Other results deal with large sets missing S ⁎ S ⁎ whose elements have algebraic properties, such as being right cancelable and generating free semigroups in βS

    Algebraic and topological equivalences in the Stone-Čech compactification of a discrete semigroup

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    AbstractWe consider the Stone-Čech compactification βS of a countably infinite discrete commutative semigroup S. We show that, under a certain condition satisfied by all cancellative semigroups S, the minimal right ideals of βS will belong to 2c homeomorphism classes. We also show that the maximal groups in a given minimal left ideal will belong to 2c homeomorphism classes. The subsets of βS of the form S + e, where e denotes an idempotent, will also belong to 2c homeomorphism classes.All the left ideals of βN of the form βN + e, where e denotes a nonminimal idempotent of βN, will be different as right topological semigroups. If e denotes a nonminimal idempotent of βZ, e + βZ will be topologically and algebraically isomorphic to precisely one other principal right ideal of βZ defined by an idempotent: −e + βZ. The corresponding statement for left ideals is also valid

    Pairwise sums in colourings of the reals

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    Suppose that we have a finite colouring of R. What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with X + X (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with X + X monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums

    Topological properties of some algebraically defined subsets of βN

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    Let S be a discrete semigroup and let the Stone–Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center. We show that the closure of the set of multiplicative idempotents in β N does not meet the set of additive idempotents in β N . We also show that the following algebraically defined subsets of β N are not Borel: the set of idempotents; the smallest ideal; any semiprincipal right ideal of N ⁎ ; the set of idempotents in any left ideal; and N ⁎ + N ⁎ . We extend these results to βS, where S is an infinite countable semigroup algebraically embeddable in a compact topological group

    Recurrence in the dynamical system (X,〈Ts〉s∈S) and ideals of βS

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    A dynamical system is a pair ( X , 〈 T s 〉 s ∈ S ) , where X is a compact Hausdorff space, S is a semigroup, for each s ∈ S , T s is a continuous function from X to X , and for all s , t ∈ S , T s ∘ T t = T s t . Given a point p ∈ β S , the Stone-Čech compactification of the discrete space S , T p : X → X is defined by, for x ∈ X , T p ( x ) = p − lim s ∈ S T s ( x ) . We let β S have the operation extending the operation of S such that β S is a right topological semigroup and multiplication on the left by any point of S is continuous. Given p , q ∈ β S , T p ∘ T q = T p q , but T p is usually not continuous. Given a dynamical system ( X , 〈 T s 〉 s ∈ S ) , and a point x ∈ X , we let U ( x ) = p ∈ β S : T p ( x ) is uniformly recurrent . We show that each U ( x ) is a left ideal of β S and for any semigroup we can get a dynamical system with respect to which K ( β S ) = ⋂ x ∈ X U ( x ) and c ℓ K ( β S ) = ⋂ U ( x ) : x ∈ X and U ( x ) is closed . And we show that weak cancellation assumptions guarantee that each such U ( x ) properly contains K ( β S ) and has U ( x ) ∖ c ℓ K ( β S ) ≠ ∅

    Dark Matter and the Chemical Evolution of Irregular Galaxies

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    We present three types of chemical evolution models for irregular galaxies: closed-box with continuous star formation rates (SFRs), closed-box with bursting SFRs, and O-rich outflow with continuous SFRs. We discuss the chemical evolution of the irregular galaxies NGC 1560 and II Zw 33, and a ``typical'' irregular galaxy. The fraction of low-mass stars needed by our models is larger than that derived for the solar vicinity, but similar to that found in globular clusters. For our typical irregular galaxy we need a mass fraction of about 40% in the form of substellar objects plus non baryonic dark matter inside the Holmberg radius, in good agreement with the results derived for NGC 1560 and II Zw 33 where we do have an independent estimate of the mass fraction in non baryonic dark matter. Closed-box models are better than O-rich outflow models in explaining the C/O and Z/O observed values for our typical irregular galaxy.Comment: 14 pages, 2 figure, uses emulateapj.sty package. ApJ in press. New models were added. The order of Tables has been correcte

    A nilpotent IP polynomial multiple recurrence theorem

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    We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate

    A New Look at the Kinematics of Neutral Hydrogen in the Small Magellanic Cloud

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    We have used the latest HI observations of the Small Magellanic Cloud (SMC), obtained with the Australia Telescope Compact Array and the Parkes telescope, to re-examine the kinematics of this dwarf, irregular galaxy. A large velocity gradient is found in the HI velocity field with a significant symmetry in iso-velocity contours, suggestive of a differential rotation. A comparison of HI data with the predictions from tidal models for the SMC evolution suggests that the central region of the SMC corresponds to the central, disk- or bar-like, component left from the rotationally supported SMC disk prior to its last two encounters with the Large Magellanic Cloud. In this scenario, the velocity gradient is expected as a left-over from the original, pre-encounter, angular momentum. We have derived the HI rotation curve and the mass model for the SMC. This rotation curve rapidly rises to about 60 km/s up to the turnover radius of ~3 kpc. A stellar mass-to-light ratio of about unity is required to match the observed rotation curve, suggesting that a dark matter halo is not needed to explain the dynamics of the SMC. A set of derived kinematic parameters agrees well with the assumptions used in tidal theoretical models that led to a good reproduction of observational properties of the Magellanic System. The dynamical mass of the SMC, derived from the rotation curve, is 2.4x10^9 Msolar.Comment: To appear in ApJ, March 20 2004, 11 figure
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