Frank–Wolfe algorithm - Revision history

en>Michael Hardy: some corrections per WP:MOS

2014-12-16T18:19:17Z

some corrections per WP:MOS

← Older revision		Revision as of 20:19, 16 December 2014
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	~~{{For\|other uses of the term\|Small set (disambiguation)}}~~		Andrew Simcox is the title his mothers and fathers gave him and he completely enjoys this title. My wife and I live in Mississippi and I love each day residing here. She functions as a travel agent but quickly she'll be on her own. To climb is something she would by no means give up.<br><br>Look at my web page [http://help.ksu.edu.sa/node/65129 accurate psychic predictions]
	~~In [[combinatorics\|combinatorial]] mathematics, a '''small set''' of [[positive integer]]s~~

	~~:<math>S = \{s_0,s_1,s_2,s_3,\dots\}</math>~~

	is ~~one such that~~ the ~~[[infinite sum]]~~

	~~:<math>\frac{1}{s_0}+\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}+\cdots</math>~~

	~~[[Series (mathematics)\|converges]]. A '''large set''' is one whose sum of reciprocals [[Series (mathematics)\|diverges]].~~

	~~==Examples==~~
	* The set <math>\{1,2,3,4,5,\dots\}</math> of all positive integers is known to be a large set (see [[Harmonic series (mathematics)\|Harmonic series]]), and ~~so is the set obtained from any [[arithmetic sequence]] (i.e. of the form ''an'' + ''b'' with ''a'' ≥ 0, ''b'' ≥ 1~~ and ~~''n'' = 0, 1, 2, 3, ..~~.~~) where ''a'' = 0, ''b'' = 1 give the multiset <math>\{1,1,1,\dots\}</math>~~ and ~~''a'' = 1, ''b'' = 1 give <math>\{1,2,3,4,5,\dots\}</math>.~~

	* The set of [[square number]]s is small (see [[Basel problem#The Riemann zeta function\|Basel problem]]). So is the set of [[cube number]]s, the set of 4th powers, and ~~so on~~. More generally, the set of values of a polynomial <math>a_k n^k+a_{k-1} n^{k-1}+\cdots+a_2 n^2+a_1 n+a_0</math>, ''k'' ≥ 2, ''a''<sub>''i''</sub> ≥ 0 for all ''i'' ≥ 1, ''a'~~'<sub>''k''</sub> > 0. When ''k''=1 we get an arithmetic sequence (which forms a large set.).~~

	* The set <math>\{1,2,4,8,\dots\}</math> of powers of [[2 (number)\|2]] is known to be ~~a small set, and so is the set of any [[geometric sequence]] (i.e~~. ~~of the form ''ab''<sup>''n''</sub> with ''a'' ≥ 1, ''b'' ≥ 2 and ''n'' = 0, 1, 2, 3, ...).~~

	* The set of [[prime number]]s [[Proof that the sum of the reciprocals of the primes diverges\|has been proven]] to be large. The set of [[twin prime]]s has been proven to be small (see [[Brun's constant]]).

	* The set of [[prime power]]s which are not prime (i.e. all ''p''<sup>''n''</sup> with ''n'' ≥ 2) is ~~a small set although the primes are a large set. This property is frequently used in [[analytic number theory]]. More generally, the set of [[perfect power]]s is small.~~

	* The set of numbers whose [[decimal]] representations exclude ''7'' (or any digit one prefers) is small. ~~That is, for example, the set~~

	:<~~math~~>~~\{\dots, 6, 8, \dots, 16, 18, \dots, 66, 68, 69, 80, \dots \}~~<~~/math~~>

	~~: is small. (This has been generalized to other [[Numeral system\|bases]] as well.) See [[Kempner series]].~~

	~~==Properties==~~
	* A [[union (set theory)\|union]] of finitely many small sets is small, as the sum of two [[convergent series]] is a convergent series. A union of infinitely many small sets is either a small set (e.g. the sets of ''p''<sup>2</sup>, ''p''<sup>3</sup>, ''p''<sup>4</sup>, ... where ''p'' is prime) or a large set (e.g. the sets <math>\{n^2 + k : n > 0 \}</math> for ''k'' > 0). Also, a large set [[complement (set theory)\|minus]] a small set is still large. A large set minus a large set is either a small set (e.g. the set of all prime powers ''p''<sup>''n''</sup> with ''n'' ≥ 1 minus the set of all primes) or a large set (e.g. the set of all positive integers minus the set of all positive even numbers). In set theoretic terminology, the small sets form an [[ideal (set theory)\|ideal]].

	* The [[Müntz–Szász theorem]] is that a set <math>S=\{s_1,s_2,s_3,\dots\}</math> is large if and only if the set spanned by

	~~:<math>\{1,x^{s_1},x^{s_2},x^{s_3},\dots\} \,</math>~~

	~~: is [[dense set\|dense]] in the [[uniform norm]] topology of [[continuous function]]s on a closed interval. This is a generalization of the [[Stone–Weierstrass theorem]].~~

	~~==Open problems==~~
	~~There are many sets about which it is not known whether they are large or small.~~

	~~Not known how to identify a large set or a small set, except proving by exhaustion.~~

	[[Paul Erdős]] famously asked the [[Erdős conjecture on arithmetic progressions\|question]] of whether any set that does not contain arbitrarily long [[arithmetic progression]]s must necessarily be small. He offered a prize of $3000 for the solution to this problem, more than for any of his [[Erdős conjectures\|other conjectures]], and joked that this prize offer violated the minimum wage law.<ref name="pomerance">[[Carl Pomerance]], [http://~~www~~.~~ams.org/notices/199801/vertesi.pdf Paul Erdős, Number Theorist Extraordinaire]. (Part of the article ''The Mathematics of Paul Erdős''), in ''[[Notices of the AMS]]'', [http://www~~.~~ams~~.~~org/notices~~/~~199801~~/~~index.html January, 1998].</ref> This question is still open.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* A. D. Wadhwa (1975). An interesting subseries of the harmonic series. ''American Mathematical Monthly'' '''82''' (9) 931–933. {{JSTOR\|2318503}}

	~~[[Category:Combinatorics]]~~
	~~[[Category:Integer sequences]]~~
	~~[[Category:Mathematical series]~~]

190.105.4.72: link to phil wolfe

2013-12-15T16:51:18Z

link to phil wolfe

← Older revision		Revision as of 18:51, 15 December 2013
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	~~Oscar~~ is ~~how he~~'s ~~called~~ and ~~he completely loves this name~~. ~~One~~ of ~~the things she enjoys most~~ is to ~~read comics~~ and ~~she~~'ll be ~~starting some thing else alongside~~ with it. ~~Minnesota~~ is ~~exactly~~ where he'~~s been residing~~ for ~~years~~. ~~Hiring~~ is ~~her working day job now~~ and ~~she will not change it whenever quickly.~~<br><br>~~Have~~ a ~~look at my page;~~ [http://www.~~videokeren~~.~~com~~/~~user~~/~~BFlockhar at home std testing~~]		{{For\|other uses of the term\|Small set (disambiguation)}}
			In [[combinatorics\|combinatorial]] mathematics, a '''small set''' of [[positive integer]]s

			:<math>S = \{s_0,s_1,s_2,s_3,\dots\}</math>

			is one such that the [[infinite sum]]

			:<math>\frac{1}{s_0}+\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}+\cdots</math>

			[[Series (mathematics)\|converges]]. A '''large set''' is one whose sum of reciprocals [[Series (mathematics)\|diverges]].

			==Examples==
			* The set <math>\{1,2,3,4,5,\dots\}</math> of all positive integers is known to be a large set (see [[Harmonic series (mathematics)\|Harmonic series]]), and so is the set obtained from any [[arithmetic sequence]] (i.e. of the form ''an'' + ''b'' with ''a'' ≥ 0, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) where ''a'' = 0, ''b'' = 1 give the multiset <math>\{1,1,1,\dots\}</math> and ''a'' = 1, ''b'' = 1 give <math>\{1,2,3,4,5,\dots\}</math>.

			* The set of [[square number]]s is small (see [[Basel problem#The Riemann zeta function\|Basel problem]]). So is the set of [[cube number]]s, the set of 4th powers, and so on. More generally, the set of values of a polynomial <math>a_k n^k+a_{k-1} n^{k-1}+\cdots+a_2 n^2+a_1 n+a_0</math>, ''k'' ≥ 2, ''a''<sub>''i''</sub> ≥ 0 for all ''i'' ≥ 1, ''a''<sub>''k''</sub> > 0. When ''k''=1 we get an arithmetic sequence (which forms a large set.).

			* The set <math>\{1,2,4,8,\dots\}</math> of powers of [[2 (number)\|2]] is known to be a small set, and so is the set of any [[geometric sequence]] (i.e. of the form ''ab''<sup>''n''</sub> with ''a'' ≥ 1, ''b'' ≥ 2 and ''n'' = 0, 1, 2, 3, ...).

			* The set of [[prime number]]s [[Proof that the sum of the reciprocals of the primes diverges\|has been proven]] to be large. The set of [[twin prime]]s has been proven to be small (see [[Brun's constant]]).

			* The set of [[prime power]]s which are not prime (i.e. all ''p''<sup>''n''</sup> with ''n'' ≥ 2) is a small set although the primes are a large set. This property is frequently used in [[analytic number theory]]. More generally, the set of [[perfect power]]s is small.

			* The set of numbers whose [[decimal]] representations exclude ''7'' (or any digit one prefers) is small. That is, for example, the set

			:<math>\{\dots, 6, 8, \dots, 16, 18, \dots, 66, 68, 69, 80, \dots \}</math>

			: is small. (This has been generalized to other [[Numeral system\|bases]] as well.) See [[Kempner series]].

			==Properties==
			* A [[union (set theory)\|union]] of finitely many small sets is small, as the sum of two [[convergent series]] is a convergent series. A union of infinitely many small sets is either a small set (e.g. the sets of ''p''<sup>2</sup>, ''p''<sup>3</sup>, ''p''<sup>4</sup>, ... where ''p'' is prime) or a large set (e.g. the sets <math>\{n^2 + k : n > 0 \}</math> for ''k'' > 0). Also, a large set [[complement (set theory)\|minus]] a small set is still large. A large set minus a large set is either a small set (e.g. the set of all prime powers ''p''<sup>''n''</sup> with ''n'' ≥ 1 minus the set of all primes) or a large set (e.g. the set of all positive integers minus the set of all positive even numbers). In set theoretic terminology, the small sets form an [[ideal (set theory)\|ideal]].

			* The [[Müntz–Szász theorem]] is that a set <math>S=\{s_1,s_2,s_3,\dots\}</math> is large if and only if the set spanned by

			:<math>\{1,x^{s_1},x^{s_2},x^{s_3},\dots\} \,</math>

			: is [[dense set\|dense]] in the [[uniform norm]] topology of [[continuous function]]s on a closed interval. This is a generalization of the [[Stone–Weierstrass theorem]].

			==Open problems==
			There are many sets about which it is not known whether they are large or small.

			Not known how to identify a large set or a small set, except proving by exhaustion.

			[[Paul Erdős]] famously asked the [[Erdős conjecture on arithmetic progressions\|question]] of whether any set that does not contain arbitrarily long [[arithmetic progression]]s must necessarily be small. He offered a prize of $3000 for the solution to this problem, more than for any of his [[Erdős conjectures\|other conjectures]], and joked that this prize offer violated the minimum wage law.<ref name="pomerance">[[Carl Pomerance]], [http://www.ams.org/notices/199801/vertesi.pdf Paul Erdős, Number Theorist Extraordinaire]. (Part of the article ''The Mathematics of Paul Erdős''), in ''[[Notices of the AMS]]'', [http://www.ams.org/notices/199801/index.html January, 1998].</ref> This question is still open.

			==Notes==
			{{reflist}}

			==References==
			* A. D. Wadhwa (1975). An interesting subseries of the harmonic series. ''American Mathematical Monthly'' '''82''' (9) 931–933. {{JSTOR\|2318503}}

			[[Category:Combinatorics]]
			[[Category:Integer sequences]]
			[[Category:Mathematical series]]

138.15.20.179: /* References */

2012-01-05T21:26:16Z

References

New page

Oscar is how he's called and he completely loves this name. One of the things she enjoys most is to read comics and she'll be starting some thing else alongside with it. Minnesota is exactly where he's been residing for years. Hiring is her working day job now and she will not change it whenever quickly.<br><br>Have a look at my page; [http://www.videokeren.com/user/BFlockhar at home std testing]