130.230.181.211: Earlier change that changed the sequence start from 0 instead of 1 didn't change the next element to 1

2013-01-15T09:06:39Z

Earlier change that changed the sequence start from 0 instead of 1 didn't change the next element to 1

← Older revision		Revision as of 11:06, 15 January 2013
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	~~Hello~~, ~~my name~~ is ~~Andrew~~ and ~~my wife doesn't like~~ it ~~at all~~. ~~To perform lacross~~ is ~~something he would by no means give up~~. ~~I am an invoicing officer~~ and I'~~ll be promoted quickly~~. ~~For~~ a ~~whilst I~~'~~ve been~~ in ~~Alaska but I will have to move~~ in ~~a yr or two~~.<br><br>~~My weblog~~ ... ~~free psychic readings~~ ([~~https~~://www.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 www~~.~~machlitim~~.org.il])		In [[mathematics]], '''piecewise syndeticity''' is a notion of largeness of subsets of the [[natural number]]s.

			A set <math>S \sub \mathbb{N}</math> is called ''piecewise syndetic'' if there exists a finite subset ''G'' of <math>\mathbb{N}</math> such that for every finite subset ''F'' of <math>\mathbb{N}</math> there exists an <math>x \in \mathbb{N}</math> such that

			:<math>x+F \subset \bigcup_{n \in G} (S-n)</math>

			where <math>S-n = \{m \in \mathbb{N}: m+n \in S \}</math>. Equivalently, ''S'' is piecewise syndetic if there are arbitrarily long intervals of <math>\mathbb{N}</math> where the gaps in ''S'' are bounded by some constant ''b''.

			== Properties ==

			* A set is piecewise syndetic if and only if it is the intersection of a [[syndetic set]] and a [[thick set]].

			* If ''S'' is piecewise syndetic then ''S'' contains arbitrarily long arithmetic progressions.

			* A set ''S'' is piecewise syndetic if and only if there exists some ultrafilter ''U'' which contains ''S'' and ''U'' is in the smallest two-sided ideal of <math>\beta \mathbb{N}</math>, the [[Stone–Čech compactification]] of the natural numbers.

			* [[partition regular\| Partition regularity]]: if <math>S</math> is piecewise syndetic and <math>S = C_1 \cup C_2 \cup ... \cup C_n</math>, then for some <math>i \leq n</math>, <math>C_i</math> contains a piecewise syndetic set. (Brown, 1968)

			* If ''A'' and ''B'' are subsets of <math>\mathbb{N}</math>, and ''A'' and ''B'' have positive [[natural density\| upper Banach density]], then <math>A+B=\{a+b:a \in A, b \in B\}</math> is piecewise syndetic<ref>R. Jin, [http://jinr.people.cofc.edu/research/banach.pdf Nonstandard Methods For Upper Banach Density Problems], ''Journal of Number Theory'' '''91''', (2001), 20-38</math>.</ref>

			== Other Notions of Largeness ==

			There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers:

			* [[Cofiniteness]]
			* [[IP set]]
			* member of a nonprincipal [[ultrafilter]]
			* positive [[upper density]]
			* [[syndetic set]]
			* [[thick set]]

			== See also ==
			*[[Ergodic Ramsey theory]]

			== Notes ==
			{{reflist}}

			== References ==
			* J. McLeod, "[http://www.mtholyoke.edu/~jmcleod/somenotionsofsize.pdf Some Notions of Size in Partial Semigroups]" ''Topology Proceedings'' '''25''' (2000), 317-332
			* [[Vitaly Bergelson]], "[http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf Minimal Idempotents and Ergodic Ramsey Theory]", ''Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310'', Cambridge Univ. Press, Cambridge, (2003)
			* [[Vitaly Bergelson]], N. Hindman, "[http://members.aol.com/nhfiles2/pdf/large.pdf Partition regular structures contained in large sets are abundant]", ''J. Comb. Theory (Series A)'' '''93''' (2001), 18-36
			* T. Brown, "[http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102971066 An interesting combinatorial method in the theory of locally finite semigroups]", ''Pacific J. Math.'' '''36''', no. 2 (1971), 285–289.

			[[Category:Semigroup theory]]
			[[Category:Ergodic theory]]
			[[Category:Ramsey theory]]
			[[Category:Combinatorics]]

en>Marc van Leeuwen: /* Equivalence to primitive recursion */ modifications to this section corresponding to previous edit

2012-01-30T12:54:33Z

Equivalence to primitive recursion: modifications to this section corresponding to previous edit

New page

Hello, my name is Andrew and my wife doesn't like it at all. To perform lacross is something he would by no means give up. I am an invoicing officer and I'll be promoted quickly. For a whilst I've been in Alaska but I will have to move in a yr or two.<br><br>My weblog ... free psychic readings ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 www.machlitim.org.il])

Course-of-values recursion - Revision history

130.230.181.211: Earlier change that changed the sequence start from 0 instead of 1 didn't change the next element to 1

en>Marc van Leeuwen: /* Equivalence to primitive recursion */ modifications to this section corresponding to previous edit