A. H. Lightstone - Revision history

en>Rjwilmsi: /* Other research */Added 2 dois to journal cites using AWB (10104)

2014-05-09T09:50:16Z

Other research: Added 2 dois to journal cites using AWB (10104)

← Older revision		Revision as of 11:50, 9 May 2014
Line 1:		Line 1:
	~~The '''Erdős–Turán conjecture''' is an old unsolved problem in [[additive number theory]] (not~~ to ~~be confused with [[Erdős conjecture on arithmetic progressions]]) posed by Paul Erdős and Pál Turán in 1941.~~		Got nothing to write about myself at all.<br>Great to be a part of this site.<br>I really wish I'm useful at all<br><br>Check out my web page: [http://blogtrekhoedep.wordpress.com/ tre khoe dep]

	~~==History==~~

	~~The conjecture was made jointly by [[Paul Erdős]] and [[Pál Turán]] in~~.<~~ref~~>~~{{Cite journal \|last=Erdős \|first=Paul. \|last2=Turán\|first2=Pál \|year=1941 \|title=On~~ a ~~problem of Sidon in additive number theory, and on some related problems \|journal=Journal~~ of ~~the London Mathematical Society \|volume=16 \|pages=212–216 \|doi=10.1112/jlms/s1-16.4.212}}</ref> In the original paper, they state~~

	~~"(2) If <math>f(n) > 0 </math> for <math>n > n_0 </math>, then <math> \varlimsup_{n \rightarrow \infty} f(n) = \infty </math>"~~

	Here <math>f(n)</math> is the number of ways one can write the natural number <math>n</math> as the sum of two (not necessarily distinct) elements of <math>B</math>. If <math>f(n)</math> is always positive for sufficiently large <math>n</math>, then <math>B</math> is called an additive basis (of order 2).<ref name="TaoVu13">{{cite book \|authorlink=Terence Tao \|first=T. \|last=Tao \|first2=V. \|last2=Vu \|title=Additive Combinatorics \|location=New York \|publisher=Cambridge University Press \|year=2006 \|isbn=0-521-85386-9 \|page=13 }}</ref> This problem has attracted significant attention<ref name="TaoVu13" /> but remains unsolved.

	~~In 1964, Erdős published a multiplicative version~~ this ~~conjecture. See source :~~

	*P. Erdõs: On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261

	~~==Progress==~~

	~~While the conjecture remains unsolved, there have been significant advance on the problem. First, we express the problem in modern language~~. ~~For a given subset <math>B \subset \mathbb{N}~~ <~~/math~~>~~, we define its~~ '~~'representation function'' <math>r_B(n) = \#\{(a_1, a_2) \in B^2 \| a_1 + a_2 = n \}</math>. Then the conjecture states that if <math> r_B(n) > 0 </math> for~~ all <~~math>n</math> sufficiently large, then <math> \limsup_{n \rightarrow \infty} r_B(n) = \infty </math>.~~

	~~More generally, for any <math>h \in \mathbb{N}</math> and subset <math~~>~~B \subset \mathbb{N}~~ <~~/math~~>, we can define the <math>h</math> representation function as <math>r_{B,h}(n) = \#\{(a_1, \cdots, a_h) \in B^h \| a_1 + \cdots + a_h = n \}</math>. We say that <math>B</math> is an additive basis of order <math>h</math> if <math>r_{B,h}(n) > 0 </math> for all <math> n </math> sufficiently large. One can see from an elementary argument that if <math>B</math> is an additive basis of order <math>h</math>, then

	~~<math> \displaystyle n \leq \sum_{m=1}^n r_{B,h}(m) \leq \|B \cap [1,n]\|^h </math>~~

	~~So we obtain the lower bound <math> n^{1/h} \leq \|B \cap [1,n]\| </math>.~~

	~~The original conjecture spawned as Erdős and Turán sought a partial answer to Sidon's problem (see: [[Sidon sequence]]). Later, Erdős set~~ out ~~to answer the following question posed by Sidon~~: ~~how close to the lower bound <math> \|B \cap~~ [1,n]\| \geq n^{1/h} </math> can an additive basis <math> B </math> of order <math> h </math> get? This question was answered positively in the case <math>h=2</math> by Erdős in 1956.<ref>{{cite journal \|first=P. \|last=Erdős \|title=Problems and results in additive number theory \|journal=Colloque sur le Theorie des Nombres \|year=1956 \|volume= \|issue= \|pages=127–137 \|doi= }}</ref> Erdős proved that there exists an additive basis <math>B</math> of order 2 and constants <math>c_1, c_2 > 0 </math> such that <math>c_1 \log n \leq r_B(n) \leq c_2 \log n </math> for all <math>n </math> sufficiently large. In particular, this implies that there exists an additive basis <math>B</math> such that <math>r_B(n) = n^{1/2 + o(1)} </math>, which is essentially best possible. This motivated Erdős to make the following conjecture

	~~If <math>B</math> is an additive basis of order <math>h</math>, then <math> \limsup_{n \rightarrow \infty} r_B(n)/\log n > 0.</math>~~

	In 1986, [[Eduard Wirsing]] proved that a large class of additive bases, including the [[prime numbers]], contains a subset that is an additive basis but significantly thinner than the original.<ref>{{Cite journal \|last = Wirsing \|first=Eduard \| year = 1986 \| title = Thin subbases \|journal=Analysis \| volume=6 \|issue= \| pages= 285–308 }}</ref> In 1990, Erdős and Tetalli extended Erdős's 1956 result to bases of arbitrary order.<ref>{{Cite journal \|last=Erdős \|first=Paul. \|last2=Tetalli\|first2=Prasad \|year=1990 \|title=Representations of integers as the sum of <math>k</math> terms\|journal=Random Structures Algorithms\|volume=1 \|issue = 3 \|pages=245–261 \|doi=10.1002/rsa.3240010302 }}</ref> In 2000, [[Van H. Vu\|V. Vu]] proved that thin subbases exist in the Waring bases using the [[Hardy–Littlewood circle method]] and his polynomial concentration results.<ref>{{Cite journal \|last = Vu \| first=Van \| year = 2000 \| title = On a refinement of Waring's problem \|journal=Duke Mathematical Journal \|volume=105 \| issue = 1 \|pages= 107–134 \|doi=10.1215/S0012-7094-00-10516-9 }}</ref> In 2006, Borwein, Choi, and Chu proved that for all additive bases <math>B</math>, <math>f(n)</math> eventually exceeds 7.<ref>{{Cite journal \|last = Borwein \| first=Peter \| last2 = Choi \| first2=Stephen \|last3=Chu \|first3=Frank\|year=2006\|title = An old conjecture of Erdős–Turán on additive bases \|journal=Mathematics of Computation \|volume=75 \|issue= \|pages=475–484 \|doi= }}</ref>
	~~<ref>{{cite book \|last1= Xiao \|first1= Stanley Yao \|title= On the Erdős–Turán conjecture and related results\|url=~~ http://~~hdl~~.~~handle~~.~~net/10012~~/~~6150 \|degree= masters \|year = 2011 }}</ref>~~

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Erdos-Turan Conjecture On Additive Bases}}~~
	~~[[Category:Additive number theory]~~]

en>Trappist the monk: /* Decimal hyperreals */Fix CS1 deprecated date parameter errors (test) using AWB

2014-02-02T01:59:21Z

Decimal hyperreals: Fix CS1 deprecated date parameter errors (test) using AWB

← Older revision		Revision as of 03:59, 2 February 2014
Line 1:		Line 1:
	~~Spyderco Gayle Bradley Folder Knife - That~~ is an ~~impressive, yet easy folding knife. The reason~~ [~~http://Istoriya.Sumy.ua8080/index.php/Best_Pocket_Knife_For_A_Farmer it's pricey~~] ~~is simply due~~ to ~~the fabric it’s made~~ with. The ~~blade is~~ a ~~hollow-floor satin completed Crucible Metal CPM-M4~~. ~~If you happen to’re uncertain what which means, it refers [http:~~//~~www~~.~~thebestpocketknifereviews~~.~~com~~/~~becker-knives-review~~/ ~~Becker Knives] to the metal type~~, m4 is ~~very powerful in opposition to any kind~~ of ~~impression – making it~~ one ~~sturdy knife~~. ~~The Crucible model of the blade~~ is ~~a process that works alongside the m4 in making a superfine~~, ~~steel grain construction making a past impressive impact resistance even more spectacular~~. ~~$152~~.46.<br><br>~~Let’s start with the plain~~, ~~hand to claw fight, deep within the forest with~~ a ~~bear~~. ~~Of course that is completely a state of affairs you will probably end up in some day, so you definitely want a high quality further sharp pocket knife – so you'll be able to later tell~~ the ~~story~~ of ~~how you fought a bear and gained with nothing but your small uninteresting pocket knife~~, ~~your crafty, and your lightning quick reflexes~~. ~~Jack knife~~. ~~A jack knife has a simple hinge at one end~~, ~~and will~~ have ~~multiple blade~~. ~~The jack knife is fashionable among hunters~~, ~~fishermen, and campers~~.<br><br>~~This is the place BestPocketKnifeToday.com comes into play. We~~'~~re here that can assist you choose the proper pocket knife to your needs and provide you with some options of our private~~. ~~There may be pile of information about pocket knives on~~ the ~~web which is strewn everywhere~~. ~~Now we have now gathered all the data you need~~ in ~~a single place~~ and ~~made it easy for you. Rest assured~~ we ~~KNOW pocket knives and hope you have~~ the ~~benefit of what the location has to offer~~. ~~Help me resolve which knife to buy!~~<br><br>~~One thing~~ is ~~evident – Boker makes some merely beautiful knives with completely extraordinary designs. The Boker Plus Subcom has~~ an ~~intriguing design and some impressive specs~~. ~~It’s only 2.6 inches lengthy when closed and has~~ an ~~AUS-8 chrome steel blade which~~ is ~~partially serrated and just shy~~ of ~~two inches in length~~. The ~~whole thing weighs only about two~~ and a ~~half ounces so most of~~ the ~~time you gained’t know you’re even carrying it. The standard is~~ [~~http:~~//~~ner.vse.cz80~~/~~wiki~~/~~index.php~~/~~Best_Pocket_Knife_For_Gardeners outstanding] and also you won’t see any wobble on the blade when locked~~ in ~~place~~. ~~Oh,~~ and ~~it is available~~ in ~~a plain blade titanium mannequin too! SOG Entry Card~~ 2.~~zero.~~<br><br>~~After I bought this knife I was anticipating to not prefer it~~, ~~but I discovered it for an excellent value~~. ~~What stunned me~~ is ~~that I instantly preferred the knife~~. ~~The lengthy smooth blade~~, ~~the deal with~~ that ~~slot in my hand perfectly~~, ~~everything seemed great. I’d not fear one bit if this were~~ the ~~knife I used to be stuck within the woods with~~, ~~although it isn’t~~ a ~~large knife. The unlucky half~~ is ~~that I found it too bulky to hold in any pants lighter in weight~~ than ~~jeans — which for me is a non-starter~~.<br><br>~~Once I look at the Bantam~~, ~~I typically see myself being nine or ten years previous in front~~ of the ~~“Jiffy,” a convenience retailer a number~~ of ~~miles from our neighborhood that we'd frequent~~. ~~I see myself using this knife to open packages of candy and cookies~~, ~~then~~ using the ~~bottle opener on an old fashioned glass bottle of Coca-Cola—a snapshot of Americana if there ever was one. I feel~~ [~~http://www~~.~~thebestpocketknifereviews~~.~~com~~/~~becker~~-~~knives~~-~~review~~/ ~~Esee Ethan Becker Cooking Knives Review] it’s a good newbie for youngsters round 7 or 8 years previous. As always~~, ~~the token tweezers~~ and ~~toothpick are there. Don’t fear. They promote replacements. I’ve misplaced lots in my time on this Earth. $18.50~~<br><br>~~Particularly I wanted to check 1~~) ~~the ability~~ of ~~the 9Cr19MoV Stainless to take a sharp edge 2) the sting retention~~ of the new metal 3) the built-in Carbide sharpener to determine its effectiveness, and likewise to see how straightforward the metal is to re-sharpen four) the finger choil to see how it would possibly have an effect on batoning and ~~featherstick-making 5) the feel [~~http://~~www~~.~~knife-depot~~.~~com~~/~~pocket~~-~~knives/ pocket knifes~~] of the knife with the brand new full tang construction whereas utilizing it out within the subject, and 6) the improved survival whistle. The metal is a big improvement in all facets, including edge-holding, ease of sharpening, and the ability to take a sharper edge than the original model.		The '''Erdős–Turán conjecture''' is an old unsolved problem in [[additive number theory]] (not to be confused with [[Erdős conjecture on arithmetic progressions]]) posed by Paul Erdős and Pál Turán in 1941.

			==History==

			The conjecture was made jointly by [[Paul Erdős]] and [[Pál Turán]] in.<ref>{{Cite journal \|last=Erdős \|first=Paul. \|last2=Turán\|first2=Pál \|year=1941 \|title=On a problem of Sidon in additive number theory, and on some related problems \|journal=Journal of the London Mathematical Society \|volume=16 \|pages=212–216 \|doi=10.1112/jlms/s1-16.4.212}}</ref> In the original paper, they state

			"(2) If <math>f(n) > 0 </math> for <math>n > n_0 </math>, then <math> \varlimsup_{n \rightarrow \infty} f(n) = \infty </math>"

			Here <math>f(n)</math> is the number of ways one can write the natural number <math>n</math> as the sum of two (not necessarily distinct) elements of <math>B</math>. If <math>f(n)</math> is always positive for sufficiently large <math>n</math>, then <math>B</math> is called an additive basis (of order 2).<ref name="TaoVu13">{{cite book \|authorlink=Terence Tao \|first=T. \|last=Tao \|first2=V. \|last2=Vu \|title=Additive Combinatorics \|location=New York \|publisher=Cambridge University Press \|year=2006 \|isbn=0-521-85386-9 \|page=13 }}</ref> This problem has attracted significant attention<ref name="TaoVu13" /> but remains unsolved.

			In 1964, Erdős published a multiplicative version this conjecture. See source :

			*P. Erdõs: On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261

			==Progress==

			While the conjecture remains unsolved, there have been significant advance on the problem. First, we express the problem in modern language. For a given subset <math>B \subset \mathbb{N} </math>, we define its ''representation function'' <math>r_B(n) = \#\{(a_1, a_2) \in B^2 \| a_1 + a_2 = n \}</math>. Then the conjecture states that if <math> r_B(n) > 0 </math> for all <math>n</math> sufficiently large, then <math> \limsup_{n \rightarrow \infty} r_B(n) = \infty </math>.

			More generally, for any <math>h \in \mathbb{N}</math> and subset <math>B \subset \mathbb{N} </math>, we can define the <math>h</math> representation function as <math>r_{B,h}(n) = \#\{(a_1, \cdots, a_h) \in B^h \| a_1 + \cdots + a_h = n \}</math>. We say that <math>B</math> is an additive basis of order <math>h</math> if <math>r_{B,h}(n) > 0 </math> for all <math> n </math> sufficiently large. One can see from an elementary argument that if <math>B</math> is an additive basis of order <math>h</math>, then

			<math> \displaystyle n \leq \sum_{m=1}^n r_{B,h}(m) \leq \|B \cap [1,n]\|^h </math>

			So we obtain the lower bound <math> n^{1/h} \leq \|B \cap [1,n]\| </math>.

			The original conjecture spawned as Erdős and Turán sought a partial answer to Sidon's problem (see: [[Sidon sequence]]). Later, Erdős set out to answer the following question posed by Sidon: how close to the lower bound <math> \|B \cap [1,n]\| \geq n^{1/h} </math> can an additive basis <math> B </math> of order <math> h </math> get? This question was answered positively in the case <math>h=2</math> by Erdős in 1956.<ref>{{cite journal \|first=P. \|last=Erdős \|title=Problems and results in additive number theory \|journal=Colloque sur le Theorie des Nombres \|year=1956 \|volume= \|issue= \|pages=127–137 \|doi= }}</ref> Erdős proved that there exists an additive basis <math>B</math> of order 2 and constants <math>c_1, c_2 > 0 </math> such that <math>c_1 \log n \leq r_B(n) \leq c_2 \log n </math> for all <math>n </math> sufficiently large. In particular, this implies that there exists an additive basis <math>B</math> such that <math>r_B(n) = n^{1/2 + o(1)} </math>, which is essentially best possible. This motivated Erdős to make the following conjecture

			If <math>B</math> is an additive basis of order <math>h</math>, then <math> \limsup_{n \rightarrow \infty} r_B(n)/\log n > 0.</math>

			In 1986, [[Eduard Wirsing]] proved that a large class of additive bases, including the [[prime numbers]], contains a subset that is an additive basis but significantly thinner than the original.<ref>{{Cite journal \|last = Wirsing \|first=Eduard \| year = 1986 \| title = Thin subbases \|journal=Analysis \| volume=6 \|issue= \| pages= 285–308 }}</ref> In 1990, Erdős and Tetalli extended Erdős's 1956 result to bases of arbitrary order.<ref>{{Cite journal \|last=Erdős \|first=Paul. \|last2=Tetalli\|first2=Prasad \|year=1990 \|title=Representations of integers as the sum of <math>k</math> terms\|journal=Random Structures Algorithms\|volume=1 \|issue = 3 \|pages=245–261 \|doi=10.1002/rsa.3240010302 }}</ref> In 2000, [[Van H. Vu\|V. Vu]] proved that thin subbases exist in the Waring bases using the [[Hardy–Littlewood circle method]] and his polynomial concentration results.<ref>{{Cite journal \|last = Vu \| first=Van \| year = 2000 \| title = On a refinement of Waring's problem \|journal=Duke Mathematical Journal \|volume=105 \| issue = 1 \|pages= 107–134 \|doi=10.1215/S0012-7094-00-10516-9 }}</ref> In 2006, Borwein, Choi, and Chu proved that for all additive bases <math>B</math>, <math>f(n)</math> eventually exceeds 7.<ref>{{Cite journal \|last = Borwein \| first=Peter \| last2 = Choi \| first2=Stephen \|last3=Chu \|first3=Frank\|year=2006\|title = An old conjecture of Erdős–Turán on additive bases \|journal=Mathematics of Computation \|volume=75 \|issue= \|pages=475–484 \|doi= }}</ref>
			<ref>{{cite book \|last1= Xiao \|first1= Stanley Yao \|title= On the Erdős–Turán conjecture and related results\|url= http://hdl.handle.net/10012/6150 \|degree= masters \|year = 2011 }}</ref>

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Erdos-Turan Conjecture On Additive Bases}}
			[[Category:Additive number theory]]

en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-04T06:46:13Z

ISBNs (Build KC)

New page

Spyderco Gayle Bradley Folder Knife - That is an impressive, yet easy folding knife. The reason [http://Istoriya.Sumy.ua8080/index.php/Best_Pocket_Knife_For_A_Farmer it's pricey] is simply due to the fabric it’s made with. The blade is a hollow-floor satin completed Crucible Metal CPM-M4. If you happen to’re uncertain what which means, it refers [http://www.thebestpocketknifereviews.com/becker-knives-review/ Becker Knives] to the metal type, m4 is very powerful in opposition to any kind of impression – making it one sturdy knife. The Crucible model of the blade is a process that works alongside the m4 in making a superfine, steel grain construction making a past impressive impact resistance even more spectacular. $152.46. Let’s start with the plain, hand to claw fight, deep within the forest with a bear. Of course that is completely a state of affairs you will probably end up in some day, so you definitely want a high quality further sharp pocket knife – so you'll be able to later tell the story of how you fought a bear and gained with nothing but your small uninteresting pocket knife, your crafty, and your lightning quick reflexes. Jack knife. A jack knife has a simple hinge at one end, and will have multiple blade. The jack knife is fashionable among hunters, fishermen, and campers. This is the place BestPocketKnifeToday.com comes into play. We're here that can assist you choose the proper pocket knife to your needs and provide you with some options of our private. There may be pile of information about pocket knives on the web which is strewn everywhere. Now we have now gathered all the data you need in a single place and made it easy for you. Rest assured we KNOW pocket knives and hope you have the benefit of what the location has to offer. Help me resolve which knife to buy! One thing is evident – Boker makes some merely beautiful knives with completely extraordinary designs. The Boker Plus Subcom has an intriguing design and some impressive specs. It’s only 2.6 inches lengthy when closed and has an AUS-8 chrome steel blade which is partially serrated and just shy of two inches in length. The whole thing weighs only about two and a half ounces so most of the time you gained’t know you’re even carrying it. The standard is [http://ner.vse.cz80/wiki/index.php/Best_Pocket_Knife_For_Gardeners outstanding] and also you won’t see any wobble on the blade when locked in place. Oh, and it is available in a plain blade titanium mannequin too! SOG Entry Card 2.zero. After I bought this knife I was anticipating to not prefer it, but I discovered it for an excellent value. What stunned me is that I instantly preferred the knife. The lengthy smooth blade, the deal with that slot in my hand perfectly, everything seemed great. I’d not fear one bit if this were the knife I used to be stuck within the woods with, although it isn’t a large knife. The unlucky half is that I found it too bulky to hold in any pants lighter in weight than jeans — which for me is a non-starter. Once I look at the Bantam, I typically see myself being nine or ten years previous in front of the “Jiffy,” a convenience retailer a number of miles from our neighborhood that we'd frequent. I see myself using this knife to open packages of candy and cookies, then using the bottle opener on an old fashioned glass bottle of Coca-Cola—a snapshot of Americana if there ever was one. I feel [http://www.thebestpocketknifereviews.com/becker-knives-review/ Esee Ethan Becker Cooking Knives Review] it’s a good newbie for youngsters round 7 or 8 years previous. As always, the token tweezers and toothpick are there. Don’t fear. They promote replacements. I’ve misplaced lots in my time on this Earth. $18.50 Particularly I wanted to check 1) the ability of the 9Cr19MoV Stainless to take a sharp edge 2) the sting retention of the new metal 3) the built-in Carbide sharpener to determine its effectiveness, and likewise to see how straightforward the metal is to re-sharpen four) the finger choil to see how it would possibly have an effect on batoning and featherstick-making 5) the feel [http://www.knife-depot.com/pocket-knives/ pocket knifes] of the knife with the brand new full tang construction whereas utilizing it out within the subject, and 6) the improved survival whistle. The metal is a big improvement in all facets, including edge-holding, ease of sharpening, and the ability to take a sharper edge than the original model.