Representation theory of the Lorentz group: Difference between revisions

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[[Image:No-three-in-line.svg|thumb|A set of 20 points in a 10 × 10 grid, with no three points in a line.]]
== A word. 'Hung then ==
In [[mathematics]], in the area of [[discrete geometry]], the '''no-three-in-line'''-problem, introduced by [[Henry Dudeney]] in 1917, asks for the maximum number of points that can be placed in the ''n'' × ''n'' grid so that no three points are [[collinear]]. This number is at most 2''n'', since if 2''n'' + 1 points are placed in the grid some row will contain three points.


==Lower bounds==
A word. 'Hung then, after a virtual assistant automatically translated into the [http://www.nrcil.net/fancybox/lib/rakuten_LV_125.html ルイヴィトン アズール バッグ] lingua [http://www.nrcil.net/fancybox/lib/rakuten_LV_120.html ルイヴィトン買取価格] franca of the universe.<br><br>opposite him, a white robe stood [http://www.nrcil.net/fancybox/lib/rakuten_LV_123.html ルイヴィトン中古通販] a tall thin man, the man holding a two-edged sword, watching the flood: '! I did not think the killing fields of battle space in which you can [http://www.nrcil.net/fancybox/lib/rakuten_LV_54.html ルイヴィトン 財布 新作] hit an opponent.'<br><br>'again.' Man Di soon.<br><br>Phew!<br><br>stature [http://www.nrcil.net/fancybox/lib/rakuten_LV_94.html ルイヴィトン タイガ 財布] a lightning pediment, direct flood forced close, but in the hands of the sword as if spirituality in general seems ready to attack across the strange arc to flood many parts of the body.<br><br>'Ha ha [http://www.nrcil.net/fancybox/lib/rakuten_LV_125.html ルイヴィトン アズール バッグ] ha [http://www.nrcil.net/fancybox/lib/rakuten_LV_93.html 中古ルイヴィトン] ha ......' Hung happy for a laugh.<br><br>gunpoint guns in the hands of a roll suddenly, as if the waves rolled thousands of opponents to sweep the general directly, while it shares marksmanship among the dark mood directly into the field, as if the dark terrible whirlpool swallowed directly to the front of the white gowns young swordsman, sword in the hands [http://www.nrcil.net/fancybox/lib/rakuten_LV_51.html ルイヴィトン エピ 財布] of young man in white swordsman extremely neat, with a little bit like the electro-optic
[[Paul Erdős]] (in {{harvnb|Roth|1951}}) observed that, when ''n'' is a [[prime number]], the set of ''n'' grid points (''i'', ''i''<sup>2</sup> mod ''n''), for 0 ≤ ''i'' &lt; ''n'', contains no three collinear points. When ''n'' is not prime, one can perform this construction for a ''p'' × ''p'' grid contained in the ''n'' × ''n'' grid, where ''p'' is the largest prime that is at most ''n''As a consequence, for any ε and any sufficiently large ''n'', one can place
相关的主题文章:
:<math>(1 - \epsilon)n</math>
  <ul>
points in the ''n'' × ''n'' grid with no three points collinear.
 
  <li>[http://www.panierscadeauxinternationaux.com/cgi-bin/store/index.cgi http://www.panierscadeauxinternationaux.com/cgi-bin/store/index.cgi]</li>
 
  <li>[http://tool.qycn.com/deadlink/?url=www.gpthanhhoa.org/cgi-bin/fpg.cgi http://tool.qycn.com/deadlink/?url=www.gpthanhhoa.org/cgi-bin/fpg.cgi]</li>
 
  <li>[http://www.wormwoodreview.com/cgi-bin/guestbook/guestbook.cgi http://www.wormwoodreview.com/cgi-bin/guestbook/guestbook.cgi]</li>
 
</ul>


Erdős' bound has been improved subsequently: {{harvtxt|Hall|Jackson|Sudbery|Wild|1975}} show that, when ''n''/2 is prime, one can obtain a solution with 3(''n'' - 2)/2 points by placing points on the [[hyperbola]] ''xy'' ≡ ''k'' (mod ''n''/2) for a suitable ''k''. Again, for arbitrary ''n'' one can perform this construction for a prime near ''n''/2 to obtain a solution with
== 'seconded ==
:(3/2 − ε)''n''
points.


==Conjectured upper bounds==
However, compliance with respect [http://www.nrcil.net/fancybox/lib/rakuten_LV_115.html ルイヴィトン お財布] to absorb Feng Luo. '<br><br>'seconded!'<br><br>'seconded!'<br>Some [http://www.nrcil.net/fancybox/lib/rakuten_LV_12.html ルイヴィトン 中古] strong presence<br>suddenly hesitated slightly after agreed.<br><br>'Hung League, absorb a full member, [http://www.nrcil.net/fancybox/lib/rakuten_LV_55.html 中古ルイヴィトン] must [http://www.nrcil.net/fancybox/lib/rakuten_LV_109.html ルイヴィトン ミニバッグ] be really strong and many ethnic groups! If we can rely on some of the treasures, special means to kill elementary champion, was able to prove his strength., but ...... I [http://www.nrcil.net/fancybox/lib/rakuten_LV_53.html ルイヴィトン 財布 メンズ] am worried about is that it is possible is his primary protectors kill champion. protectors and that they did not [http://www.nrcil.net/fancybox/lib/rakuten_LV_100.html ルイヴィトン 中古品] register in the military system, the military [http://www.nrcil.net/fancybox/lib/rakuten_LV_140.html ルイヴィトン バック] system that automatically finds ...... Luo Feng servants, military medals natural body count in Feng Luo. 'a loose with a tentacle-like braids of aquamarine life shrill voice, 'So, I applied for transfer to see the video of that war.'<br><br>'tune to see the fight video, can prove everything.'<br><br>'industry is Venerable, out of the fight video of it.' Many are looking to the strong presence of tall purple armor armored humans.<br><br>'Luo [http://www.nrcil.net/fancybox/lib/rakuten_LV_12.html ルイヴィトン オンライン] Feng view all fighting video
{{harvtxt|Guy|Kelly|1968}} conjectured that for large ''n'' one cannot do better than ''c n'' with
相关的主题文章:
:<math>c = \sqrt[3]{\frac{2\pi^2}{3}} \approx 1.874.</math>
<ul>
{{harvtxt|Pegg, Jr.|2005}} noted that Gabor Ellmann found, in March 2004, an error in the original paper of Guy and Kelly's heuristic reasoning,
 
which if corrected, results in
  <li>[http://www.proprat.com/cgi-bin/ppsite2/proprat_website.cgi http://www.proprat.com/cgi-bin/ppsite2/proprat_website.cgi]</li>
:<math>c = \frac{\pi}{\sqrt 3} \approx 1.814.</math>
 
  <li>[http://ensembleada.com/publicity_next12/sunbbs.cgi http://ensembleada.com/publicity_next12/sunbbs.cgi]</li>
 
  <li>[http://www.digital-dany.com/cgi/guestbook/guestbook.cgi http://www.digital-dany.com/cgi/guestbook/guestbook.cgi]</li>
 
</ul>


==Applications==
== 「低音の惑星 ==
The [[Heilbronn triangle problem]] asks for the placement of ''n'' points in a unit square that maximizes the area of the smallest triangle formed by three of the points. By applying Erdős' construction of a set of grid points with no three collinear points, one can find a placement in which the smallest triangle has area
:<math>\frac{1-\epsilon}{2n^2}.</math>


==Generalizations==
イルカは遠い遠い星を見ることができます [http://www.nrcil.net/fancybox/lib/rakuten_LV_60.html ルイヴィトン 値段]。<br><br>「低音の惑星。「羅風水の目は天に最も近い遠くにある惑星に落ちた暗赤色の全体であるが、それは、惑星の白、濃い赤色の異質、ガス貯留白い雲です主な惑星の色、それが羅風水のの最初の区間である「低音の惑星。 [http://www.nrcil.net/fancybox/lib/rakuten_LV_73.html ルイヴィトン シューズ] '<br><br>「殿下は、どのようにするつもりですか? [http://www.nrcil.net/fancybox/lib/rakuten_LV_111.html ルイヴィトン 柄] 'ディランは風水羅に見えた。<br><br>'ディランは、私は個人的にこの作業を行う必要があり、介入しない最初から最後まで、覚えています。「羅風水を行う方法については、私の心がされている」、目は寒さのタッチを渡して、遠くの惑星を見て沈没判断に。 [http://www.nrcil.net/fancybox/lib/rakuten_LV_107.html ルイヴィトン ケース] '<br><br>いわゆる最高得点タスク、羅風水は混乱を感じ、混乱していたしましょう​​ [http://www.nrcil.net/fancybox/lib/rakuten_LV_133.html ルイヴィトン 新作 2014]。<br><br>しかし、単にLuofengはほぼ元の文字の最初の仮想世界ファム会員企業が変だ、考え出したいくつかの時間を考慮して、血に飢え、寒さ、穏やかな、最終的にはいくつかの特別な才能がある
A noncollinear placement of ''n'' points can also be interpreted as a [[graph drawing]] of the [[complete graph]] in such a way that, although edges cross, no edge passes through a vertex. Erdős' construction above can be generalized to show that every ''n''-vertex ''k''-colorable graph has such a drawing in a ''O''(''n'') &times; ''O''(''k'') grid ({{harvtxt|Wood|2005}}).
相关的主题文章:
<ul>
 
  <li>[http://cgi.www5b.biglobe.ne.jp/~kaneda/cgi-bin/ita2/ita_honey_b_b_s5.cgi http://cgi.www5b.biglobe.ne.jp/~kaneda/cgi-bin/ita2/ita_honey_b_b_s5.cgi]</li>
 
  <li>[http://t0321.biz/mtos/mt-search.cgi http://t0321.biz/mtos/mt-search.cgi]</li>
 
  <li>[http://people.lis.illinois.edu/~unsworth/courses/bestsellers/search.cgi http://people.lis.illinois.edu/~unsworth/courses/bestsellers/search.cgi]</li>
 
</ul>


Non-collinear sets of points in the three-dimensional grid were considered by {{harvtxt|Pór and Wood|2007}}. They proved that the maximum number of points in the ''n'' × ''n'' × ''n'' grid with no three points collinear is <math>\Theta(n^2)</math>.
== ただ、マスターからの注文に耳を傾け、銀ガード、1ストリーム ==
Similarly to Erdős's 2D construction, this can be accomplished by using points
(''x'', ''y'', ''x''<sup>2</sup> + ''y''<sup>2</sup>) mod ''p'', where ''p'' is a prime congruent to 3 mod 4.
One can also consider graph drawings in the three-dimensional grid. Here the non-collinearity condition means that a vertex should not lie on a non-adjacent edge, but it is normal to work with the stronger requirement that no two edges cross ({{harvtxt|Thiele|Tóth|1998}}; {{harvtxt|Dujmović|Morin|Wood|2005}}; {{harvtxt|Di Giacomo|Liotta|Meijer|2005}}).


==Small values of ''n''==
ガードの特別な製品。シルバーガードのこの流れは、主原料は知能システムを含む、多くの貴重な「フロー銀 [http://www.nrcil.net/fancybox/lib/rakuten_LV_76.html ルイヴィトン コレクション] '金属'黒い金属イットリウム」を超えている」馬場Taは、叫んだ [http://www.nrcil.net/fancybox/lib/rakuten_LV_82.html ショルダーバッグ ルイヴィトン] '。通常は実在の人物が好き!話しのように振る舞うことができます。 '<br><br>「強いパンチがそれらにヒットする場合でも、ボディ構造は、銀、体内の水分の流れであるため、また、彼らは簡単に安心しインパクトになります! [http://www.nrcil.net/fancybox/lib/rakuten_LV_44.html ルイヴィトン 財布 マルチカラー] '<br><br>'だから、彼らは非常に良いボディーガードです [http://www.nrcil.net/fancybox/lib/rakuten_LV_50.html ルイヴィトン ジッピーオーガナイザー]!'<br><br>」など暗殺など、彼らは物理的にQudangを完全に行うことができます。 [http://www.nrcil.net/fancybox/lib/rakuten_LV_66.html ルイヴィトン 中古 バッグ] '<br><br>「致死性が強すぎではありませんが、彼らが絡まっていますが、宇宙の強いレベルが出るには少し時間を費やす必要があります!最高のボディーガードである。「馬場Taは言った、 [http://www.nrcil.net/fancybox/lib/rakuten_LV_23.html ルイヴィトン タイガ] 'それはロボットと、「銀のガードを流れ [http://www.nrcil.net/fancybox/lib/rakuten_LV_17.html ルイヴィトン キャップ] '同じではありません。ただ、マスターからの注文に耳を傾け、銀ガード、1ストリーム。ただ、人間の生命力のような、彼らは普通の生活を維持する必要がある2、エネルギー、彼らのためにエネルギーを。 [http://www.nrcil.net/fancybox/lib/rakuten_LV_2.html ルイヴィトン2014新作] '<br><br>'OVER
For ''n'' ≤ 46, it is known that 2''n'' points may be placed with no three in a line. The numbers of solutions (not counting reflections and rotations as distinct) for small ''n'' = 2, 3, ..., are
相关的主题文章:
:[[1 (number)|1]], [[1 (number)|1]], [[4 (number)|4]], [[5 (number)|5]], [[11 (number)|11]], [[22 (number)|22]], [[57 (number)|57]], [[51 (number)|51]], [[156 (number)|156]], [[158 (number)|158]], 566, 499, 1366, ... {{OEIS|id=A000769}}.
  <ul>
 
    
==References==
   <li>[http://www.cacenglish.com/index.php?item/create_form/1 http://www.cacenglish.com/index.php?item/create_form/1]</li>
* {{citation
    
  | last = Dudeney
  <li>[http://gempakz.org/index.php?item/create_form/1 http://gempakz.org/index.php?item/create_form/1]</li>
  | first = Henry
    
  | authorlink = Henry Dudeney
  <li>[http://www.thefaulkingtruth.com/blog/blog.cgi http://www.thefaulkingtruth.com/blog/blog.cgi]</li>
  | title = Amusements in Mathematics
 
  | year = 1917
</ul>
  | location = Edinburgh
  | publisher = Nelson}}
* {{citation
  | title = Computing Straight-line 3D Grid Drawings of Graphs in Linear Volume
  | journal = Computational Geometry: Theory and Applications
  | volume = 32
  | issue = 1
  | year = 2005
  | pages = 26&ndash;58
  | last1 = Di Giacomo
  | first1 = Emilio
  | last2 = Liotta
  | first2 = Giuseppe
  | last3 = Meijer
  | first3 = Henk
  | doi = 10.1016/j.comgeo.2004.11.003}}
* {{citation
  | last1 = Dujmović
  | first1 = Vida
  | last2 = Morin
  | first2 = Pat
  | last3 = Wood
  | first3 = David R.
  | title = Layout of Graphs with Bounded Tree-Width
  | year = 2005
  | journal = [[SIAM Journal on Computing]]
  | volume = 34
  | issue = 3
  | pages = 553&ndash;579
  | doi = 10.1137/S0097539702416141}}
* {{cite journal
  | ref=harv
  | first1= Stefan
  | last1 = Felsner
  | first2 = Giuseppe
  | last2 = Liotta
  | first3 = Stephen K.  
  | last3 = Wismath
  | title = Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions
  | journal = [[Journal of Graph Algorithms and Applications]]
  | year = 2003
  | volume = 7
  | issue = 4
  | pages = 363–398
  | url=http://jgaa.info/accepted/2003/Felsner+2003.7.4.pdf}}
* {{cite journal
  | ref=harv
  | last = Flammenkamp
  | first = Achim
  | title = Progress in the no-three-in-line problem
  | journal = [[Journal of Combinatorial Theory]] | series = Series A
  | volume = 60
  | year = 1992
  | issue = 2
  | pages = 305&ndash;311
  | doi = 10.1016/0097-3165(92)90012-J}}
* {{cite journal
  | ref=harv
  | last = Flammenkamp
  | first = Achim
  | title = Progress in the no-three-in-line problem, II
  | journal = [[Journal of Combinatorial Theory]] | series = Series A
  | volume = 81
  | year = 1998
  | issue = 1
  | pages = 108&ndash;113
  | doi = 10.1006/jcta.1997.2829}}
* {{cite journal
  | ref=harv
  | doi = 10.4153/CMB-1968-062-3
  | first1 = R. K. | last1 = Guy    | author1-link = Richard K. Guy
  | first2 = P. A. | last2 = Kelly
  | title = The no-three-in-line problem
  | journal = Canadian Mathematical Bulletin
  | volume = 11
  | issue = 0
  | year = 1968
  | pages = 527&ndash;531
|id={{MathSciNet|0238765}}}}
* {{cite journal
  | ref=harv
  | last1 = Hall    | first1 = R. R.
  | last2 = Jackson | first2 = T. H.
   | last3 = Sudbery | first3 = A.
  | last4 = Wild   | first4 = K.
  | title = Some advances in the no-three-in-line problem
  | journal = [[Journal of Combinatorial Theory]] | series = Series A
  | volume = 18
  | year = 1975
  | issue = 3
  | pages = 336&ndash;341
  | doi = 10.1016/0097-3165(75)90043-6}}
* {{cite conference
  | ref=harv
  |first1=Hanno
  |last1=Lefmann
  |title=No l Grid-Points in spaces of small dimension
  |booktitle=Algorithmic Aspects in Information and Management, 4th International Conference, AAIM 2008, Shanghai, China, June 23-25, 2008, Proceedings
  |series=Lecture Notes in Computer Science | publisher = Springer-Verlag
  |volume=5034
  |year=2008
  |pages=259&ndash;270
  |doi=10.1007/978-3-540-68880-8_25}}
* {{cite conference
  | ref=harv
  | first1 = János  | last1 = Pach    | author1-link = János Pach
  | first2 = Torsten | last2 = Thiele
  | first3 = Géza    | last3 = Tóth
  | title = Three-dimensional grid drawings of graphs
  | date = 1998
  | booktitle = [[International Symposium on Graph Drawing|Graph Drawing, 5th Int. Symp., GD '97]]
  | series = Lecture Notes in Computer Science | volume= 1353 | publisher= Springer-Verlag
  | pages = 47&ndash;51
  |doi =10.1007/3-540-63938-1_49}}
* {{cite web
  | ref=harv
  | last1 = Pegg, Jr. | first1 = Ed
  | title=Math Games: Chessboard Tasks
  | url=http://www.mathpuzzle.com/MAA/36-Chessboard%20Tasks/mathgames_04_11_05.html
   | date=April 11, 2005
  | accessdate=June 25, 2012}}
* {{cite journal
  | ref=harv
  | first1 = Attila  | last1 = Pór
  | first2 = David R. | last2 = Wood
  | title = No-three-in-line-in-3D
  | journal = [[Algorithmica]]
  | year = 2007
  | doi = 10.1007/s00453-006-0158-9
  | volume = 47
  | issue = 4
  | pages = 481}}
* {{ Citation
  | last = Roth
  | first = K. F.
  | title = On a problem of Heilbronn
  | journal = [[Journal of the London Mathematical Society]]
  | volume = 26
  | year = 1951
  | issue = 3
  | pages = 198&ndash;204
  | doi = 10.1112/jlms/s1-26.3.198}}
* {{cite journal
   | ref=harv
  | first = David R.  | last = Wood
  | title = Grid drawings of ''k''-colourable graphs
  | journal = Computational Geometry: Theory and Applications
  | volume = 30
  | issue = 1
  | year = 2005
  | pages = 25&ndash;28
  | doi = 10.1016/j.comgeo.2004.06.001}}
 
==External links==
* {{cite web | author = Flammenkamp, Achim | url = http://wwwhomes.uni-bielefeld.de/achim/no3in/readme.html | title = The No-Three-in-Line Problem}}
 
* {{mathworld | title = No-Three-in-a-Line-Problem | urlname = No-Three-in-a-Line-Problem}}
 
[[Category:Combinatorics]]
[[Category:Lattice points]]
[[Category:Conjectures]]
[[Category:Mathematical problems]]

Revision as of 12:03, 1 March 2014

A word. 'Hung then

A word. 'Hung then, after a virtual assistant automatically translated into the ルイヴィトン アズール バッグ lingua ルイヴィトン買取価格 franca of the universe.

opposite him, a white robe stood ルイヴィトン中古通販 a tall thin man, the man holding a two-edged sword, watching the flood: '! I did not think the killing fields of battle space in which you can ルイヴィトン 財布 新作 hit an opponent.'

'again.' Man Di soon.

Phew!

stature ルイヴィトン タイガ 財布 a lightning pediment, direct flood forced close, but in the hands of the sword as if spirituality in general seems ready to attack across the strange arc to flood many parts of the body.

'Ha ha ルイヴィトン アズール バッグ ha 中古ルイヴィトン ha ......' Hung happy for a laugh.

gunpoint guns in the hands of a roll suddenly, as if the waves rolled thousands of opponents to sweep the general directly, while it shares marksmanship among the dark mood directly into the field, as if the dark terrible whirlpool swallowed directly to the front of the white gowns young swordsman, sword in the hands ルイヴィトン エピ 財布 of young man in white swordsman extremely neat, with a little bit like the electro-optic 相关的主题文章:

'seconded

However, compliance with respect ルイヴィトン お財布 to absorb Feng Luo. '

'seconded!'

'seconded!'
Some ルイヴィトン 中古 strong presence
suddenly hesitated slightly after agreed.

'Hung League, absorb a full member, 中古ルイヴィトン must ルイヴィトン ミニバッグ be really strong and many ethnic groups! If we can rely on some of the treasures, special means to kill elementary champion, was able to prove his strength., but ...... I ルイヴィトン 財布 メンズ am worried about is that it is possible is his primary protectors kill champion. protectors and that they did not ルイヴィトン 中古品 register in the military system, the military ルイヴィトン バック system that automatically finds ...... Luo Feng servants, military medals natural body count in Feng Luo. 'a loose with a tentacle-like braids of aquamarine life shrill voice, 'So, I applied for transfer to see the video of that war.'

'tune to see the fight video, can prove everything.'

'industry is Venerable, out of the fight video of it.' Many are looking to the strong presence of tall purple armor armored humans.

'Luo ルイヴィトン オンライン Feng view all fighting video 相关的主题文章:

「低音の惑星

イルカは遠い遠い星を見ることができます ルイヴィトン 値段

「低音の惑星。「羅風水の目は天に最も近い遠くにある惑星に落ちた暗赤色の全体であるが、それは、惑星の白、濃い赤色の異質、ガス貯留白い雲です主な惑星の色、それが羅風水のの最初の区間である「低音の惑星。 ルイヴィトン シューズ '

「殿下は、どのようにするつもりですか? ルイヴィトン 柄 'ディランは風水羅に見えた。

'ディランは、私は個人的にこの作業を行う必要があり、介入しない最初から最後まで、覚えています。「羅風水を行う方法については、私の心がされている」、目は寒さのタッチを渡して、遠くの惑星を見て沈没判断に。 ルイヴィトン ケース '

いわゆる最高得点タスク、羅風水は混乱を感じ、混乱していたしましょう​​ ルイヴィトン 新作 2014

しかし、単にLuofengはほぼ元の文字の最初の仮想世界ファム会員企業が変だ、考え出したいくつかの時間を考慮して、血に飢え、寒さ、穏やかな、最終的にはいくつかの特別な才能がある 相关的主题文章:

ただ、マスターからの注文に耳を傾け、銀ガード、1ストリーム

ガードの特別な製品。シルバーガードのこの流れは、主原料は知能システムを含む、多くの貴重な「フロー銀 ルイヴィトン コレクション '金属'黒い金属イットリウム」を超えている」馬場Taは、叫んだ ショルダーバッグ ルイヴィトン '。通常は実在の人物が好き!話しのように振る舞うことができます。 '

「強いパンチがそれらにヒットする場合でも、ボディ構造は、銀、体内の水分の流れであるため、また、彼らは簡単に安心しインパクトになります! ルイヴィトン 財布 マルチカラー '

'だから、彼らは非常に良いボディーガードです ルイヴィトン ジッピーオーガナイザー!'

」など暗殺など、彼らは物理的にQudangを完全に行うことができます。 ルイヴィトン 中古 バッグ '

「致死性が強すぎではありませんが、彼らが絡まっていますが、宇宙の強いレベルが出るには少し時間を費やす必要があります!最高のボディーガードである。「馬場Taは言った、 ルイヴィトン タイガ 'それはロボットと、「銀のガードを流れ ルイヴィトン キャップ '同じではありません。ただ、マスターからの注文に耳を傾け、銀ガード、1ストリーム。ただ、人間の生命力のような、彼らは普通の生活を維持する必要がある2、エネルギー、彼らのためにエネルギーを。 ルイヴィトン2014新作 '

'OVER 相关的主题文章: