402 research outputs found

### Effectiveness of Hindman's theorem for bounded sums

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 $\mathsf{HT}^{\leq n}_k$ denote the
assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite
set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F
\subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a
computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite
computable set $X$ such that all nonempty sums of at most $2$ elements of $X$
have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable
in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in
$\mathsf{RCA}_0$. We also show that there is a computable instance of
$\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this
result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in
$\mathsf{RCA}_0$

### The scarcity of products in ÎČS â S

Let S be a discrete semigroup and let the StoneâÄech compactification ÎČS of S have the operation extending that of S which makes ÎČS a right topological semigroup with S contained in its topological center. Let S â = ÎČ S â S . Algebraically, the set of products S â S â tends to be rather large, since it often contains the smallest ideal of ÎČS. We establish here sufficient conditions involving mild cancellation assumptions and assumptions about the cardinality of S for S â S â to be topologically small, that is for S â S â to be nowhere dense in S â , or at least for S â â S â S â to be dense in S â . 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 â S â whose elements have algebraic properties, such as being right cancelable and generating free semigroups in ÎČS

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

AbstractWe consider the Stone-Äech compactification ÎČS 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 ÎČS 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 ÎČS of the form S + e, where e denotes an idempotent, will also belong to 2c homeomorphism classes.All the left ideals of ÎČN of the form ÎČN + e, where e denotes a nonminimal idempotent of ÎČN, will be different as right topological semigroups. If e denotes a nonminimal idempotent of ÎČZ, e + ÎČZ will be topologically and algebraically isomorphic to precisely one other principal right ideal of ÎČZ defined by an idempotent: âe + ÎČZ. The corresponding statement for left ideals is also valid

### Pairwise sums in colourings of the reals

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 ÎČN

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

### Recurrence in the dynamical system (X,ăTsăsâS) and ideals of ÎČS

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

### Dark Matter and the Chemical Evolution of Irregular Galaxies

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

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

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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