Z-buffering - Revision history

en>Jodosma: copyedit: derivated→derived

2014-10-06T19:44:46Z

copyedit: derivated→derived

← Older revision		Revision as of 21:44, 6 October 2014
Line 1:		Line 1:
	In [[~~number theory~~]~~], '''Goldbach~~'s ~~weak conjecture''', also known as the '''odd Goldbach conjecture''', the '''ternary Goldbach problem''', or the '''3-primes problem''', states that~~:		Dalton is what's written for his birth certificate but he never really treasured that name. The [http://Www.Bing.com/search?q=precious+hobby&form=MSNNWS&mkt=en-us&pq=precious+hobby precious hobby] for him and his kids is you will need but he's been claiming on new things a short time ago. Auditing is even his primary income proceeds from. Massachusetts is where your lover and his [http://En.search.wordpress.com/?q=wife+live wife live]. He's not godd at design but consider want to check our website: http://prometeu.net<br><br>

	~~: Every [[odd number]] greater than 5 can be expressed as the sum of three [[prime number\|primes]]. (A prime may be used more than once in the same sum.)~~		Here is my web site; [http://prometeu.net clash of clans hack]

	This [[conjecture]] is called "weak" because if [[Goldbach's conjecture\|Goldbach's strong conjecture]] (concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7.) In 2013, [[Harald Helfgott]] claimed to have fully proved the conjecture for all odd integers greater than 5 (rather than the much larger <math>e^{3100}\approx 2 \times 10^{1346}</math>, implied by previous results).

	~~Some state the conjecture as:~~
	~~:Every odd number greater than 7 can be expressed as the sum of three odd primes.<ref>{{MathWorld\|title=Goldbach Conjecture\|id=GoldbachConjecture}}</ref>~~
	~~This version excludes 7 = 2+2+3 because this requires the even prime 2. Helfgott's claim covers both versions of the conjecture.~~

	~~== Status ==~~
	Earlier partial and/or conditional results on the conjecture include the following: In 1923, [[G. H. Hardy\|Hardy]] and [[John Edensor Littlewood\|Littlewood]] showed that, assuming the [[generalized Riemann hypothesis]], the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, [[Ivan Matveevich Vinogradov]] eliminated the dependency on the generalised Riemann hypothesis and proved directly (see [[Vinogradov's theorem]]) that all [[sufficiently large]] odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective [[Siegel–Walfisz theorem]], did not give a bound for "sufficiently large"; his student K. Borozdin proved that 3<sup>3<sup>15</sup></sup> is large enough.<ref>Golomb gives the date of Borozdin's proof as 1956; in contrast, Tao states that it was "soon after" Vinogradov's 1937 proof. {{citation\|title=The invincible primes\|first=Solomon W.\|last=Golomb\|authorlink=Solomon W. Golomb\|journal=The Sciences\|volume=25\|issue=2\|pages=50–57\|year=1985\|doi=10.1002/j.2326-1951.1985.tb02782.x}}; {{citation\|doi=10.1007/978-3-642-19533-4_1\|contribution=Structure and Randomness in the Prime Numbers\|first=Terence\|last=Tao\|authorlink=Terence Tao\|pages=1–7\|year=2011\|title=An Invitation to Mathematics: From Competitions to Research\|editor1-first=Dierk\|editor1-last=Schleicher\|editor2-first=Malte\|editor2-last=Lackmann\|publisher=Springer}}, footnote 7, [http://~~books.google.com/books?id=9TATfteVeVYC&pg=PA1 p. 1].</ref> This number has 6,846,169 decimal digits, so checking every number under this figure would be completely unfeasible~~.

	In 1997, [[Jean-Marc Deshouillers\|Deshouillers]], Effinger, [[Herman te Riele\|te Riele]] and Zinoviev published a result showing<ref>{{cite journal\|title=A complete Vinogradov 3-primes theorem under the Riemann hypothesis\|author=Deshouillers, Effinger, Te Riele and Zinoviev\|journal=Electronic Research Announcements of the American Mathematical Society\|volume=3\|pages=99–104\|year=1997\|url=http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-6762-97-00031-0.pdf\|format=PDF\|doi=10.1090/S1079-6762-97-00031-0\|issue=15}}</ref> that the [[generalized Riemann hypothesis]] implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 10<sup>20</sup> with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time.<ref>{{cite journal\|title=Checking the odd Goldbach Conjecture up to 10<sup>20</sup>\|author=Yannick Saouter\|journal=Mathematics of Computation\|volume=67\|pages=863–866\|year=1998\|url=http://www.ams.org/journals/mcom/1998-67-222/S0025-5718-98-00928-4/S0025-5718-98-00928-4.pdf\|format=PDF\|doi=10.1090/S0025-5718-98-00928-4 \|issue=222}}</ref>

	[[Olivier Ramaré]] in 1995 showed that every even number ''n'' ≥ 4 is in fact the sum of at most six primes, from which it follows that every odd number ''n'' ≥ 5 is the sum of at most seven primes. [[Leszek Kaniecki]] showed every odd integer is a sum of at most five primes, under the [[Riemann Hypothesis]].<ref>{{cite journal\|title=On Šnirelman's constant under the Riemann hypothesis\|last=Kaniecki\|first=Leszek\|journal=Acta Arithmetica\|volume=72\|year=1995\|pages=361–374\|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7246.pdf}}</ref> In 2012, [[Terence Tao]] proved this without the Riemann Hypothesis; this improves both results.<ref>{{Cite arXiv\|last=Tao \|first=Terence\|title=Every odd number greater than 1 is the sum of at most five primes \|eprint=1201.6656v4 \|year=2012 \|bibcode=2012arXiv1201.6656T \|class=math.NT}}</ref>

	In 2002, Liu Ming-Chit ([[University of Hong Kong]]) and Wang Tian-Ze lowered this threshold to approximately <math>n>e^{3100}\approx 2 \times 10^{1346}</math>. The [[exponent]] is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 10<sup>18</sup> for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.)

	In 2012 and 2013, Peruvian mathematician [[Harald Helfgott]] released a pair of papers claiming to improve major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture.<ref>{{cite arXiv \|eprint=1305.2897 \|title = Major arcs for Goldbach's theorem\|last = Helfgott\|first = H.A. \|class=math.NT \|year=2013}}</ref><ref> {{cite arXiv \|eprint=1205.5252/ \|title = Minor arcs for Goldbach's problem \|last = Helfgott\|first = H.A.\|class=math.NT \|year=2012}}</ref><ref>{{cite arXiv \|eprint=1312.7748 \|title = The ternary Goldbach conjecture is true\|last = Helfgott\|first = H.A. \|class=math.NT \|year=2013}}</ref> Here, the major arcs <math>\mathfrak M</math> is the union of intervals <math>\left (a/q-cr_0/qx,a/q+cr_0/qx\right )</math> around the rationals <math>a/q,q<r_0</math> where <math>c</math> is a contant. Minor arcs <math>\mathfrak{m}</math> are defined to be <math>\mathfrak{m}=(\mathbb R/\mathbb Z)\setminus\mathfrak{M}</math>.

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Additive number theory]]~~
	~~[[Category:Analytic number theory]]~~
	~~[[Category:Conjectures about prime numbers]]~~

	~~[[ru:Проблема Гольдбаха#Тернарная проблема Гольдбаха]~~]

149.3.96.129: /* Uses */ This Wikipedia page is not an advert for Maya, nor for Blender, and is also not about video editing. Also, the shadow mapping use of the Z buffer is not confined to scanline renderering, nor does it project z-buffer data "onto the ground".

2013-11-16T22:34:40Z

Uses: This Wikipedia page is not an advert for Maya, nor for Blender, and is also not about video editing. Also, the shadow mapping use of the Z buffer is not confined to scanline renderering, nor does it project z-buffer data "onto the ground".

← Older revision		Revision as of 00:34, 17 November 2013
Line 1:		Line 1:
	~~Michael Jackson once~~ known as ~~one of~~ the ~~top most pop singer has had~~ the ~~same~~ problem~~. Micheal~~'~~s longtime makeup artist~~, ~~Karen Kaye~~, ~~remembers how he struggled with the disease~~:~~. The abundance of genetic diseases associated with vitiligo clearly proves that there are a~~ number ~~of genes which normally govern~~ the ~~development and wellbeing~~ of the ~~melanocytes~~. ~~Cosmetics can also~~ be ~~used~~ to ~~cover up~~ the ~~white spots~~. ~~Even some renowned skin care and cosmetic products manufactures also claim~~ to ~~contain herbal oil and natural substances in their remedy.~~ <br><br>~~In people who have vitiligo~~, the ~~immune cells (bodily cells that fight against infection) attack~~ the ~~melanocytes and destroy them in addition to eliminating infections~~. ~~They claim~~ the ~~treatment plan might also help people vitiligo~~. ~~There are several theories regarding the cause~~ of ~~vitiligo but~~ the ~~actual cause is not fully known~~. ~~Novel patches also come into view over time~~ and ~~can be generalized over large portion of~~ the ~~body or localized to a meticulous part. Herbal medicines useful for vitiligo are~~: ~~Manjishtha (Rubia cordifolia)~~, ~~Saariva (Hemidesmus indicus)~~, ~~Triphala (Three fruits)~~, ~~Haridra (Curcuma longa)~~, ~~Daruharidra~~ (~~Berberis aristata~~), ~~Khadir (Acacia catechu)~~, ~~Vidanga (Embilia ribes) and Bavachi (Psoralia corylifolia)~~. <br><br>~~Some repigmentation took place in 57% - 82%~~ of ~~the handled places, in one research. That~~'s ~~why they keep creating drugs which are expensive~~, ~~of limited effectiveness, and~~ that ~~should be taken continuously for many years~~. ~~Take 1 gm from this powder and mix it with 100 ml milk~~. ~~He also emphasized that he was more than proud to be black~~ and ~~that all~~ the ~~rumors that people spread were so untrue and did more than just hurt him~~. ~~Vitiligo affects the scalp hair color, leaving spots or streaks white~~. <br><br>~~Due to unnecessary sun exposure cells~~ and ~~tissues~~ of the ~~skin becomes so dehydrated~~ that ~~they change~~ the ~~natural color of their skin~~. This ~~depigmentation of skin can be reduces~~ with ~~skin lightner products such as Benoquin made~~ of ~~real monobenzone~~. ~~On the other hand, Vitiligo can~~ also ~~occur underneath~~ the ~~arms as well as in~~ the ~~groin area~~. ~~These particular study also suggests that natural [~~http://~~discover~~-~~prague.info~~/ ~~treatment of vitiligo] can be effective~~. ~~Sometimes the best treatment for vitiligo is no treatment at all~~. <br>~~<br>The first Spanish settlement on~~ the ~~island was the town~~ of ~~Sevilla~~, which ~~was abandoned around 1524, because~~ it ~~was considered unhealthy. To use this, you should start by cleansing your skin with a product~~ that ~~suits you~~. ~~Magnesium~~ is ~~another important constituent~~ of ~~this herb~~. ~~Thus~~, ~~there are treatments for~~ the ~~condition out in~~ the ~~marketplace. Many~~ of ~~her insightful articles can be found~~ at ~~the premier online news magazine For more information on Rejuvinol~~, go to. <br><br>It is ~~a skin disorder mainly due~~ to ~~inappropriate diet combination intake regularly and formed due to disturbance of tridosas namely 'Vaat'~~, ~~'Pitta'~~ and ~~'Kapha'. For the motive~~ that ~~antibiotics can be supportive to overcome a disease~~ for ~~a specific time period, but it can't be efficient for a longer period of time~~. In ~~the early period of the disease the white spots are extremely undersized~~ and ~~limited to body parts. To control your emotions having~~ a ~~chemical substance referred~~ to ~~as monobenzylether of hydroquinone~~ and ~~also takes about 12 months~~ to ~~accomplish~~. The ~~disease often starts appearing on~~ the ~~hands~~, ~~feet or face~~, ~~and frequently pigment loss~~ is ~~progressive~~.		In [[number theory]], '''Goldbach's weak conjecture''', also known as the '''odd Goldbach conjecture''', the '''ternary Goldbach problem''', or the '''3-primes problem''', states that:

			: Every [[odd number]] greater than 5 can be expressed as the sum of three [[prime number\|primes]]. (A prime may be used more than once in the same sum.)

			This [[conjecture]] is called "weak" because if [[Goldbach's conjecture\|Goldbach's strong conjecture]] (concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7.) In 2013, [[Harald Helfgott]] claimed to have fully proved the conjecture for all odd integers greater than 5 (rather than the much larger <math>e^{3100}\approx 2 \times 10^{1346}</math>, implied by previous results).

			Some state the conjecture as:
			:Every odd number greater than 7 can be expressed as the sum of three odd primes.<ref>{{MathWorld\|title=Goldbach Conjecture\|id=GoldbachConjecture}}</ref>
			This version excludes 7 = 2+2+3 because this requires the even prime 2. Helfgott's claim covers both versions of the conjecture.

			== Status ==
			Earlier partial and/or conditional results on the conjecture include the following: In 1923, [[G. H. Hardy\|Hardy]] and [[John Edensor Littlewood\|Littlewood]] showed that, assuming the [[generalized Riemann hypothesis]], the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, [[Ivan Matveevich Vinogradov]] eliminated the dependency on the generalised Riemann hypothesis and proved directly (see [[Vinogradov's theorem]]) that all [[sufficiently large]] odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective [[Siegel–Walfisz theorem]], did not give a bound for "sufficiently large"; his student K. Borozdin proved that 3<sup>3<sup>15</sup></sup> is large enough.<ref>Golomb gives the date of Borozdin's proof as 1956; in contrast, Tao states that it was "soon after" Vinogradov's 1937 proof. {{citation\|title=The invincible primes\|first=Solomon W.\|last=Golomb\|authorlink=Solomon W. Golomb\|journal=The Sciences\|volume=25\|issue=2\|pages=50–57\|year=1985\|doi=10.1002/j.2326-1951.1985.tb02782.x}}; {{citation\|doi=10.1007/978-3-642-19533-4_1\|contribution=Structure and Randomness in the Prime Numbers\|first=Terence\|last=Tao\|authorlink=Terence Tao\|pages=1–7\|year=2011\|title=An Invitation to Mathematics: From Competitions to Research\|editor1-first=Dierk\|editor1-last=Schleicher\|editor2-first=Malte\|editor2-last=Lackmann\|publisher=Springer}}, footnote 7, [http://books.google.com/books?id=9TATfteVeVYC&pg=PA1 p. 1].</ref> This number has 6,846,169 decimal digits, so checking every number under this figure would be completely unfeasible.

			In 1997, [[Jean-Marc Deshouillers\|Deshouillers]], Effinger, [[Herman te Riele\|te Riele]] and Zinoviev published a result showing<ref>{{cite journal\|title=A complete Vinogradov 3-primes theorem under the Riemann hypothesis\|author=Deshouillers, Effinger, Te Riele and Zinoviev\|journal=Electronic Research Announcements of the American Mathematical Society\|volume=3\|pages=99–104\|year=1997\|url=http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-6762-97-00031-0.pdf\|format=PDF\|doi=10.1090/S1079-6762-97-00031-0\|issue=15}}</ref> that the [[generalized Riemann hypothesis]] implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 10<sup>20</sup> with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time.<ref>{{cite journal\|title=Checking the odd Goldbach Conjecture up to 10<sup>20</sup>\|author=Yannick Saouter\|journal=Mathematics of Computation\|volume=67\|pages=863–866\|year=1998\|url=http://www.ams.org/journals/mcom/1998-67-222/S0025-5718-98-00928-4/S0025-5718-98-00928-4.pdf\|format=PDF\|doi=10.1090/S0025-5718-98-00928-4 \|issue=222}}</ref>

			[[Olivier Ramaré]] in 1995 showed that every even number ''n'' ≥ 4 is in fact the sum of at most six primes, from which it follows that every odd number ''n'' ≥ 5 is the sum of at most seven primes. [[Leszek Kaniecki]] showed every odd integer is a sum of at most five primes, under the [[Riemann Hypothesis]].<ref>{{cite journal\|title=On Šnirelman's constant under the Riemann hypothesis\|last=Kaniecki\|first=Leszek\|journal=Acta Arithmetica\|volume=72\|year=1995\|pages=361–374\|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7246.pdf}}</ref> In 2012, [[Terence Tao]] proved this without the Riemann Hypothesis; this improves both results.<ref>{{Cite arXiv\|last=Tao \|first=Terence\|title=Every odd number greater than 1 is the sum of at most five primes \|eprint=1201.6656v4 \|year=2012 \|bibcode=2012arXiv1201.6656T \|class=math.NT}}</ref>

			In 2002, Liu Ming-Chit ([[University of Hong Kong]]) and Wang Tian-Ze lowered this threshold to approximately <math>n>e^{3100}\approx 2 \times 10^{1346}</math>. The [[exponent]] is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 10<sup>18</sup> for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.)

			In 2012 and 2013, Peruvian mathematician [[Harald Helfgott]] released a pair of papers claiming to improve major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture.<ref>{{cite arXiv \|eprint=1305.2897 \|title = Major arcs for Goldbach's theorem\|last = Helfgott\|first = H.A. \|class=math.NT \|year=2013}}</ref><ref> {{cite arXiv \|eprint=1205.5252/ \|title = Minor arcs for Goldbach's problem \|last = Helfgott\|first = H.A.\|class=math.NT \|year=2012}}</ref><ref>{{cite arXiv \|eprint=1312.7748 \|title = The ternary Goldbach conjecture is true\|last = Helfgott\|first = H.A. \|class=math.NT \|year=2013}}</ref> Here, the major arcs <math>\mathfrak M</math> is the union of intervals <math>\left (a/q-cr_0/qx,a/q+cr_0/qx\right )</math> around the rationals <math>a/q,q<r_0</math> where <math>c</math> is a contant. Minor arcs <math>\mathfrak{m}</math> are defined to be <math>\mathfrak{m}=(\mathbb R/\mathbb Z)\setminus\mathfrak{M}</math>.

			==References==
			{{reflist}}

			[[Category:Additive number theory]]
			[[Category:Analytic number theory]]
			[[Category:Conjectures about prime numbers]]

			[[ru:Проблема Гольдбаха#Тернарная проблема Гольдбаха]]

en>Wbm1058: per WP:OVERLINK, unlink hardware—it isn't helpful in this context—if a link is needed, link to the specific type of hardware which manages image depth coordinates in 3-D graphics

2012-03-29T14:23:48Z

per WP:OVERLINK, unlink hardware—it isn't helpful in this context—if a link is needed, link to the specific type of hardware which manages image depth coordinates in 3-D graphics

New page

Michael Jackson once known as one of the top most pop singer has had the same problem. Micheal's longtime makeup artist, Karen Kaye, remembers how he struggled with the disease:. The abundance of genetic diseases associated with vitiligo clearly proves that there are a number of genes which normally govern the development and wellbeing of the melanocytes. Cosmetics can also be used to cover up the white spots. Even some renowned skin care and cosmetic products manufactures also claim to contain herbal oil and natural substances in their remedy. In people who have vitiligo, the immune cells (bodily cells that fight against infection) attack the melanocytes and destroy them in addition to eliminating infections. They claim the treatment plan might also help people vitiligo. There are several theories regarding the cause of vitiligo but the actual cause is not fully known. Novel patches also come into view over time and can be generalized over large portion of the body or localized to a meticulous part. Herbal medicines useful for vitiligo are: Manjishtha (Rubia cordifolia), Saariva (Hemidesmus indicus), Triphala (Three fruits), Haridra (Curcuma longa), Daruharidra (Berberis aristata), Khadir (Acacia catechu), Vidanga (Embilia ribes) and Bavachi (Psoralia corylifolia). Some repigmentation took place in 57% - 82% of the handled places, in one research. That's why they keep creating drugs which are expensive, of limited effectiveness, and that should be taken continuously for many years. Take 1 gm from this powder and mix it with 100 ml milk. He also emphasized that he was more than proud to be black and that all the rumors that people spread were so untrue and did more than just hurt him. Vitiligo affects the scalp hair color, leaving spots or streaks white. Due to unnecessary sun exposure cells and tissues of the skin becomes so dehydrated that they change the natural color of their skin. This depigmentation of skin can be reduces with skin lightner products such as Benoquin made of real monobenzone. On the other hand, Vitiligo can also occur underneath the arms as well as in the groin area. These particular study also suggests that natural [http://discover-prague.info/ treatment of vitiligo] can be effective. Sometimes the best treatment for vitiligo is no treatment at all. The first Spanish settlement on the island was the town of Sevilla, which was abandoned around 1524, because it was considered unhealthy. To use this, you should start by cleansing your skin with a product that suits you. Magnesium is another important constituent of this herb. Thus, there are treatments for the condition out in the marketplace. Many of her insightful articles can be found at the premier online news magazine For more information on Rejuvinol, go to. It is a skin disorder mainly due to inappropriate diet combination intake regularly and formed due to disturbance of tridosas namely 'Vaat', 'Pitta' and 'Kapha'. For the motive that antibiotics can be supportive to overcome a disease for a specific time period, but it can't be efficient for a longer period of time. In the early period of the disease the white spots are extremely undersized and limited to body parts. To control your emotions having a chemical substance referred to as monobenzylether of hydroquinone and also takes about 12 months to accomplish. The disease often starts appearing on the hands, feet or face, and frequently pigment loss is progressive.