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<channel>
	<title>pingpong &amp;laquo; WordPress.com Tag Feed</title>
	<link>http://en.wordpress.com/tag/pingpong/</link>
	<description>Feed of posts on WordPress.com tagged "pingpong"</description>
	<pubDate>Thu, 24 Dec 2009 02:41:06 +0000</pubDate>

	<generator>http://en.wordpress.com/tags/</generator>
	<language>en</language>

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<title><![CDATA[Peraturan Pingpong Terbaru]]></title>
<link>http://jonoterbakar.wordpress.com/2009/12/21/peraturan-pingpong-terbaru/</link>
<pubDate>Mon, 21 Dec 2009 11:39:46 +0000</pubDate>
<dc:creator>jono terbakar</dc:creator>
<guid>http://jonoterbakar.wordpress.com/2009/12/21/peraturan-pingpong-terbaru/</guid>
<description><![CDATA[Informasi bagi para pemain pingpong diseluruh galaksi. Peraturan main pingpong yang baru sudah lahir]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/ga6zAEB9fOM&#038;rel=0&#038;fs=1&#038;showsearch=0&#038;hd=0' /><param name='allowfullscreen' value='true' /><param name='wmode' value='transparent' /><embed src='http://www.youtube.com/v/ga6zAEB9fOM&#038;rel=0&#038;fs=1&#038;showsearch=0&#038;hd=0' type='application/x-shockwave-flash' allowfullscreen='true' width='425' height='350' wmode='transparent'></embed></object></span></p>
<p>Informasi bagi para pemain pingpong diseluruh galaksi. Peraturan main pingpong yang baru sudah lahir, akikahnya menyusul ya.</p>
<p><strong>1.</strong> <em>Sistem poin bukan rally, melainkan sitem pindah bola seperti badminton. Setiap regu yang tidak berhasil mengembalikan bola ke daerah musuh akan kehilangan hak-nya untuk servis.</em></p>
<p><!--morebaca muntahan selanjutnya--><strong>2.</strong> <em>Pemenang adalah yang pertama kali mencapai poin 30.</em></p>
<p><strong>3.</strong> <em>Ada 3 pilihan bantuan yang bisa digunakan oleh sebuah tim. Ketiga pilihan itu berlaku tiap set dan akan di reset.</em></p>
<blockquote><p>a) <strong>50:50</strong> &#8212;&#8212;&#62; salah satu lawan akan menyingkir dari meja pertandingan dan tidak boleh ikut dalam permainan selama selang waktu 1 poin.<br />
b) <strong>Phone the friend / telepon sahabat</strong> &#8212;&#8211;&#62; musuh akan mengangkat telepon (walaupun dia tidak sedang ditelepon), dan diwajibkan menjauh dari meja pertandingan selama 1 poin.<br />
c) <strong>Ask The Audience</strong><br />
Boleh bertanya pada penonton. Apapun pertanyaannya minumnya teh botol sosro :p </p></blockquote>
<p><strong>4.</strong> <em>Set penentu kemenangan disebut</em> <strong>SUDDENDEATH</strong> <em>set. Maksudnya, walaupun sebuah tim sudah memenangkan 10set dari 10set jika tim itu kalah pada </em><strong>SUDDENDEATH</strong><em> set maka tim itu berhak menyandang status pecundang.</em></p>
<p>Sekian peraturan pingpong yang tidak bermutu dan sangat sesat, mohon untuk tidak diperhatikan dan dianggap sebagai kentut sajalah. Datang tidak bilang-bilang hilang meninggalkan bau yang mendalam. </p>
<p>Terimakasih</p>
<p><strong>jonoterbakar</strong></p>
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<title><![CDATA[Bangkok Dangerous]]></title>
<link>http://whataboutbrian.wordpress.com/2009/11/13/bangkok-dangerous/</link>
<pubDate>Fri, 13 Nov 2009 16:12:33 +0000</pubDate>
<dc:creator>Brian</dc:creator>
<guid>http://whataboutbrian.wordpress.com/2009/11/13/bangkok-dangerous/</guid>
<description><![CDATA[Geen zorgen, die titel is gewoon een truc om lezers te trekken! :p Alles is in orde. Ik zit nog stee]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Geen zorgen, die titel is gewoon een truc om lezers te trekken! :p<br />
Alles is in orde.</p>
<p>Ik zit nog steeds in Bangkok, was de eerste dagen zo moe van de reis en alle indrukken dat ik maar besloten heb om dagtrips te maken. Zondag ga ik naar Cambodja.</p>
<p>De afgelopen week heb ik heel veel gedaan: Tijger tempel, watervallen, Bridge over the River Kwai, olifanten rijden, sky bar (trendy bar boven op een wolkenkrabber!), thai massage, fish massage, beetje gewinkeld, PingPong Show (:O later meer), real floating market, schorpioenen, sprinkhanen en meelwormen gegeten. En nog veel meer! Heb ook veel andere reizigers ontmoet en elke avond gaan we wel iets drinken. Gezellig!</p>
<p>Ook ben ik erachter gekomen dat Tuktuk en taxi niet efficient zijn, maar de Motor taxi dat wel is, echt heel stoer! De motor gaat overal doorheen en staat nooit stil. Dus erg Bangkok dangerous <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_wink.gif' alt=';)' class='wp-smiley' />  Jan en ik hebben wel een stoere Tuktuk rit gehad, waar we de chauffeur hebben overgehaald om een wheelie te maken!! Dat was echt lachen!</p>
<p>Als het goed is doet mijn Macbook het morgen weer, dan zal ik fotos uploaden en een langere blog plaatsen. Ik moet nu gaan, want dit internet cafe is niet 24 uur per dag geopend.. Dus word eruit gezet!</p>
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<title><![CDATA[Sport: sei ori per il wheelchair di pingpong]]></title>
<link>http://nutrimente2.wordpress.com/2009/11/12/sport-sei-ori-per-il-wheelchair-di-pingpong/</link>
<pubDate>Thu, 12 Nov 2009 13:19:53 +0000</pubDate>
<dc:creator>nutrimente2</dc:creator>
<guid>http://nutrimente2.wordpress.com/2009/11/12/sport-sei-ori-per-il-wheelchair-di-pingpong/</guid>
<description><![CDATA[Sei medaglie d&#8217;oro, tre d&#8217;argento e tre di bronzo, un bottino straordinario per la nazio]]></description>
<content:encoded><![CDATA[Sei medaglie d&#8217;oro, tre d&#8217;argento e tre di bronzo, un bottino straordinario per la nazio]]></content:encoded>
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<title><![CDATA[Caricature for Ping Pong Priest]]></title>
<link>http://caricaturist.wordpress.com/2009/10/29/caricature-for-ping-pong-priest/</link>
<pubDate>Thu, 29 Oct 2009 08:11:28 +0000</pubDate>
<dc:creator>luthfimustafah</dc:creator>
<guid>http://caricaturist.wordpress.com/2009/10/29/caricature-for-ping-pong-priest/</guid>
<description><![CDATA[]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img class="alignnone size-full wp-image-1576" title="Ping-Pong-Priest-3" src="http://caricaturist.wordpress.com/files/2009/11/ping-pong-priest-3.jpg" alt="Ping-Pong-Priest-3" width="400" height="533" /></p>
</div>]]></content:encoded>
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<title><![CDATA[Watching film]]></title>
<link>http://lani0905.wordpress.com/2009/10/27/film-time/</link>
<pubDate>Tue, 27 Oct 2009 16:55:19 +0000</pubDate>
<dc:creator>lani0905</dc:creator>
<guid>http://lani0905.wordpress.com/2009/10/27/film-time/</guid>
<description><![CDATA[I&#8217;m about to watch the film Triumph of the Neards on PingPong. Let&#8217;s hope my lousy lapto]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>I&#8217;m about to watch the film <em>Triumph of the Neards </em>on PingPong.<br />
Let&#8217;s hope my lousy laptop can handle all the streaming content&#8230; There are no garanties - all technology isn&#8217;t good, unfortunately.</p>
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<item>
<title><![CDATA[Ping-Pong]]></title>
<link>http://inorbt.com/2009/10/26/ping-pong/</link>
<pubDate>Mon, 26 Oct 2009 02:18:20 +0000</pubDate>
<dc:creator>orbtblog</dc:creator>
<guid>http://inorbt.com/2009/10/26/ping-pong/</guid>
<description><![CDATA[Pois é, gostosas também jogam]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><a href="http://orbt.files.wordpress.com/2009/10/pingpong.jpg"><img title="ping pong" style="border-right:0;border-top:0;display:block;float:none;margin-left:auto;border-left:0;margin-right:auto;border-bottom:0;" height="480" alt="ping pong" src="http://orbt.files.wordpress.com/2009/10/pingpong_thumb.jpg?w=358&#038;h=480" width="358" border="0" /></a></p>
<p align="center">Pois é, gostosas também jogam </p>
</div>]]></content:encoded>
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<item>
<title><![CDATA[¿Se puede jugar a pingpong en grupo?]]></title>
<link>http://emiligene.wordpress.com/2009/09/15/%c2%bfse-puede-jugar-a-pingpong-en-grupo/</link>
<pubDate>Tue, 15 Sep 2009 21:42:54 +0000</pubDate>
<dc:creator>emiligene</dc:creator>
<guid>http://emiligene.wordpress.com/2009/09/15/%c2%bfse-puede-jugar-a-pingpong-en-grupo/</guid>
<description><![CDATA[Pues sí, y además pasárselo pipa. Basta ser niño y no ser de familia rica:]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Pues sí, y además pasárselo pipa. Basta ser niño y no ser de familia rica:</p>
<p><img src="http://emiligene.wordpress.com/files/2009/09/pingpong_grupo.jpg" alt="pingpong_grupo" title="pingpong_grupo" width="500" height="347" class="alignnone size-full wp-image-312" /></p>
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<title><![CDATA[China Finally Allows Pingpong Champ A Girlfriend]]></title>
<link>http://alindenauer.wordpress.com/2009/09/02/china-finally-allows-pingpong-champ-a-girlfriend/</link>
<pubDate>Wed, 02 Sep 2009 10:42:05 +0000</pubDate>
<dc:creator>alindenauer</dc:creator>
<guid>http://alindenauer.wordpress.com/2009/09/02/china-finally-allows-pingpong-champ-a-girlfriend/</guid>
<description><![CDATA[As the reigning table tennis world champion with two Olympic silver medals under his belt, China]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>As the reigning <span id="lw_1251881048_0" style="background:none transparent scroll repeat 0 0;cursor:hand;border-bottom:medium none;">table tennis world champion</span> with two Olympic <span id="lw_1251881048_1">silver medals</span> under his belt, <span id="lw_1251881048_2">China&#8217;s Wang Hao</span> almost had it all — except a girlfriend.</p>
<p style="text-align:center;"><img class="aligncenter" src="http://legendsrevealed.com/sports/wp-content/uploads/2009/07/wanghao.jpg" alt="" /></p>
<p>The 25-year-old was banned from dating until recently, when national team officials permitted his relationship with former national teammate, 23-year-old Peng Luyang, the government-owned China Daily reported.</p>
<p>&#8220;Both of them are old enough and it&#8217;s normal,&#8221; the newspaper quoted Peng&#8217;s coach Qiao Yunping as saying.</p>
<p>Strict control of athletes&#8217; personal lives is common in <span id="lw_1251881048_4" style="background:none transparent scroll repeat 0 0;cursor:hand;border-bottom:medium none;">China</span>&#8217;s rigid state-run sporting system, which grooms young hopefuls in specialized sports schools around the country to become gold medalists, providing them with <span id="lw_1251881048_5">intensive training</span> and free food, clothes and shelter.</p>
<p>Under the <span id="lw_1251881048_6">watchful eye</span> of team officials, star athletes are often banned from dating or marrying until a certain age, restricted in endorsement contracts and sometimes have a large percentage of their winnings taken away.</p>
<p>Athletes who date without permission risk being punished. In 2004, Wang started dating another fellow national team player, Fan Ying, and officials kicked Fan off the national team. Media reports said Wang avoided punishment at the time because his world ranking was much higher than Fan&#8217;s.</p>
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<title><![CDATA[Ping Pong 1]]></title>
<link>http://laserartillery.wordpress.com/2009/08/29/pingpong-1/</link>
<pubDate>Sat, 29 Aug 2009 08:56:03 +0000</pubDate>
<dc:creator>rekno13</dc:creator>
<guid>http://laserartillery.wordpress.com/2009/08/29/pingpong-1/</guid>
<description><![CDATA[As inspired by Tiger Beer&#8217;s Translate Ping Pong project. The idea is to get two artists to pas]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>As inspired by Tiger Beer&#8217;s Translate Ping Pong project. The idea is to get two artists to pass an art file between them, adding more layers each time.</p>
<p>Ping Pong 1&#8217;s theme: &#8220;When the world ends&#8230;&#8221;</p>
<p>Move 1, Loftis:</p>
<p><a href="http://laserartillery.wordpress.com/files/2009/08/20090823-gameone4.jpg"><img class="aligncenter size-full wp-image-18" title="20090823-gameone.jpg" src="http://laserartillery.wordpress.com/files/2009/08/20090823-gameone4.jpg" alt="20090823-gameone.jpg" width="600" height="514" /></a></p>
<p>Move 2, Hsu:</p>
<p><a href="http://laserartillery.wordpress.com/files/2009/08/20090824-gameone2.jpg"><img class="aligncenter size-full wp-image-19" title="20090824-gameone" src="http://laserartillery.wordpress.com/files/2009/08/20090824-gameone2.jpg" alt="20090824-gameone" /></a></p>
<p>Move 3, Loftis:</p>
<p><a href="http://laserartillery.wordpress.com/files/2009/08/20090827-gameone1.jpg"><img class="aligncenter size-full wp-image-20" title="20090827-gameone" src="http://laserartillery.wordpress.com/files/2009/08/20090827-gameone1.jpg" alt="20090827-gameone" /></a></p>
<p>Move 4, Hsu:</p>
<p><a href="http://laserartillery.wordpress.com/files/2009/08/20090828-gameone1.jpg"><img class="aligncenter size-full wp-image-21" title="20090828-gameone" src="http://laserartillery.wordpress.com/files/2009/08/20090828-gameone1.jpg"></a></p>
<p>Move 5, Loftis:</p>
<p><a href="http://laserartillery.wordpress.com/files/2009/08/20090829-gameone1.jpg"><img class="aligncenter size-full wp-image-22" title="20090829-gameone" src="http://laserartillery.wordpress.com/files/2009/08/20090829-gameone1.jpg" alt="20090829-gameone" /></a></p>
<p>Move 6 and final, Hsu:</p>
<p><a href="http://laserartillery.wordpress.com/files/2009/08/20090830-gameone.jpg"><img class="aligncenter size-full wp-image-27" title="20090830-gameone" src="http://laserartillery.wordpress.com/files/2009/08/20090830-gameone.jpg" alt="20090830-gameone" /></a></p>
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<title><![CDATA[Birokrasi Pingpong Berganda di Kampus UNESA (1)]]></title>
<link>http://scouteng.wordpress.com/2009/08/11/birokrasi-pingpong-berganda-di-kampus-unesa-1/</link>
<pubDate>Tue, 11 Aug 2009 16:18:49 +0000</pubDate>
<dc:creator>scouteng</dc:creator>
<guid>http://scouteng.wordpress.com/2009/08/11/birokrasi-pingpong-berganda-di-kampus-unesa-1/</guid>
<description><![CDATA[Suatu hari saya membantu salah satu keluarga saya mengurus keringanan untuk mengangsur biaya pendidi]]></description>
<content:encoded><![CDATA[Suatu hari saya membantu salah satu keluarga saya mengurus keringanan untuk mengangsur biaya pendidi]]></content:encoded>
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<title><![CDATA[6 sporturi]]></title>
<link>http://alinatoma.wordpress.com/2009/08/10/6-sporturi/</link>
<pubDate>Mon, 10 Aug 2009 12:08:50 +0000</pubDate>
<dc:creator>Alina Toma</dc:creator>
<guid>http://alinatoma.wordpress.com/2009/08/10/6-sporturi/</guid>
<description><![CDATA[Ziua de duminica a fost foarte productiva din punct de vedere sportiv. Am inceput dis de dimineata c]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Ziua de duminica a fost foarte productiva din punct de vedere sportiv.</p>
<p><a href="http://www.joacatenis.ro/category/terenuri-de-tenis/" target="_blank"><img class="alignleft size-medium wp-image-727" title="tennis_racket" src="http://alinatoma.wordpress.com/files/2009/08/tennis_racket.gif?w=300" alt="tennis_racket" width="300" height="236" /></a>Am inceput dis de dimineata cu un <strong><a href="http://www.joacatenis.ro/terenuri-de-tenis/tenis-in-crangasi/" target="_blank">tenis in Crangasi</a></strong>, la ora 9. Mi-am dat seama ca sa <strong><a href="http://www.joacatenis.ro/" target="_blank">joc tenis</a></strong> nu e asa de usor precum parea, si ca trebuie sa mai exersez mult ca sa ajung la un nivel cat de cat acceptabil. Antrenorul meu personal spune ca nu il ascult. Adica nu fac ce imi spune el. Ai rabdare, te rog!</p>
<p>Dupa aceea am plecat spre Decathlon, Vali incercand sa ma inveseleasca pentru ca am realizat ca nu ma pricep la niciun sport (cam tarziu!). Acolo ne-am plimbat cu bicicletele prin magazin, am jucat fotbal (nici la asta nu ma pricep), am jucat tenis de masa (cu buletinul poti sa iei o pereche de palete si sa testezi mesele de pingpong pe care le au la vanzare), iar la iesire erau amplasate cosuri de basket, asa ca am batut mingea putin si pe acolo.</p>
<p>Seara, cand am ajuns acasa, Vio m-a intrebat daca am chef de jogging prin Politehnica. Cu gandul la fripturile mancate sambata la aniversarea zilei de nastere a lui <a href="http://vladmihuta.wordpress.com/" target="_blank">Vlad </a>(care by the way este azi, asa ca: &#8220;La multi ani!&#8221;), am acceptat, desi era tarziu si mai degraba as fi lenevit prin camera.</p>
<p>Recensamantul a fost de 6 tipuri de sport practicate intr-o zi. Yey!</p>
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<title><![CDATA[Troublegum]]></title>
<link>http://buszon.wordpress.com/2009/07/27/troublegum/</link>
<pubDate>Mon, 27 Jul 2009 12:37:15 +0000</pubDate>
<dc:creator>buszon</dc:creator>
<guid>http://buszon.wordpress.com/2009/07/27/troublegum/</guid>
<description><![CDATA[A Therapy? &#8220;Troublegum&#8221; c. albuma sokáig nem tetszett. Olyan kis slágeres rádiózenének t]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><a href="http://en.wikipedia.org/wiki/Troublegum"><img class="alignleft size-full wp-image-382" title="TherapyTroubleGum" src="http://buszon.wordpress.com/files/2009/07/therapytroublegum.jpg" alt="TherapyTroubleGum" width="200" height="200" /></a>A Therapy? &#8220;Troublegum&#8221; c. albuma sokáig nem tetszett. Olyan kis slágeres rádiózenének tartottam, vagy nem is tudom. Semmi esetre sem passzolt a Sepulturához, illetve a keményebb vonalhoz, amit akkoriban sokat hallgattam. Aztán &#8216;94 nyarán egyszer-kétszer Pozsiéknál pingpongoztunk, és mivel ő nagyon rákattant az albumra, egész délután ez szólt&#8230; Azóta nekem is tetszik, haha!</p>
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<title><![CDATA[Ping Pong]]></title>
<link>http://lousberg.wordpress.com/2009/07/09/ping-pong/</link>
<pubDate>Thu, 09 Jul 2009 16:10:23 +0000</pubDate>
<dc:creator>lousberg</dc:creator>
<guid>http://lousberg.wordpress.com/2009/07/09/ping-pong/</guid>
<description><![CDATA[Pingback &#8211; ein lustiges Wort, das mich ein wenig an Ping Pong erinnert &#8230; &#8230; und so ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Pingback &#8211; ein lustiges Wort, das mich ein wenig an Ping Pong erinnert <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_wink.gif' alt=';-)' class='wp-smiley' />  &#8230;</p>
<p>&#8230; und so falsch liegt dieser Vergleich wohl auch gar nicht. Wikipedia hilft <a href="http://de.wikipedia.org/wiki/Pingback">hier</a> mal wieder weiter und natürlich die Kommentare auf Bibliothek2.009.</p>
<div id="attachment_174" class="wp-caption aligncenter" style="width: 160px"><img class="size-thumbnail wp-image-174 " title="ping pong Schläger" src="http://lousberg.wordpress.com/files/2009/07/ping.jpg?w=150" alt="CC: by-nc-sa by the_amanda" width="150" height="112" /><p class="wp-caption-text">CC: by-nc-sa by the_amanda</p></div>
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<title><![CDATA[Marble Maze Nokia Symbian Serie60v3]]></title>
<link>http://mobileshouse.wordpress.com/2009/06/23/marble-maze-nokia-symbian-serie60v3/</link>
<pubDate>Tue, 23 Jun 2009 20:45:46 +0000</pubDate>
<dc:creator>saboor38</dc:creator>
<guid>http://mobileshouse.wordpress.com/2009/06/23/marble-maze-nokia-symbian-serie60v3/</guid>
<description><![CDATA[Marble Maze is a labyrinth game that utilizes the orientation sensor built inside some Nokia mobile ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p style="text-align:center;"><img class="alignnone" title="maze" src="http://circles.globe.com.ph/mobile/files/2009/02/maze.jpg" alt="" width="240" height="320" /></p>
<p style="text-align:center;"><span style="position:static!important;text-decoration:underline;background-image:none!important;background-repeat:repeat!important;background-attachment:scroll!important;background-color:transparent!important;cursor:pointer!important;display:inline!important;color:#6633cc;padding-bottom:1px!important;font-size:12px;font-weight:normal;font-style:normal;font-family:Tahoma;background-position:0 50%;">Marble Maze</span><span> is a labyrinth game that utilizes the orientation sensor built inside some Nokia mobile phones (at least Nokia N95, N95 8GB, and N82). In the game you control a ball inside a labyrinth by tilting the device in your hands. Currently there are 40 different labyrinths to solve, and 3 different balls, a metal ball, a rubber ball, and a super pingpong ball. First fields are easy, but they get harder once you advance. The game requires skill, accuracy, and determination, and is highly addictive. The best time is recorded for each field, so you can also try to break <span style="position:static!important;text-decoration:underline;background-image:none!important;background-repeat:repeat!important;background-attachment:scroll!important;background-color:transparent!important;cursor:pointer!important;display:inline!important;color:#6633cc;padding-bottom:1px!important;font-size:12px;font-weight:normal;font-style:normal;font-family:Tahoma;background-position:0 50%;">the record</span> once you clear a field.</span></p>
<p style="text-align:center;"><span><!--more--><span><strong>Download link :</strong><a href="http://www.ziddu.com/download/5314056/marble_maze_MobliesHouse.blogspot.com.rar.html"> </a></span><a href="http://www.ziddu.com/download/5314056/marble_maze_MobliesHouse.blogspot.com.rar.html">Marble Maze</a></span></p>
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<title><![CDATA[Marble Maze Nokia Symbian Serie60v3]]></title>
<link>http://saboor38.wordpress.com/2009/06/23/marble-maze-nokia-symbian-serie60v3/</link>
<pubDate>Tue, 23 Jun 2009 20:45:44 +0000</pubDate>
<dc:creator>saboor38</dc:creator>
<guid>http://saboor38.wordpress.com/2009/06/23/marble-maze-nokia-symbian-serie60v3/</guid>
<description><![CDATA[Marble Maze is a labyrinth game that utilizes the orientation sensor built inside some Nokia mobile ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p style="text-align:center;"><img class="alignnone" title="maze" src="http://circles.globe.com.ph/mobile/files/2009/02/maze.jpg" alt="" width="240" height="320" /></p>
<p style="text-align:center;"><span style="position:static!important;text-decoration:underline;background-image:none!important;background-repeat:repeat!important;background-attachment:scroll!important;background-color:transparent!important;cursor:pointer!important;display:inline!important;color:#6633cc;padding-bottom:1px!important;font-size:12px;font-weight:normal;font-style:normal;font-family:Tahoma;background-position:0 50%;">Marble Maze</span><span> is a labyrinth game that utilizes the orientation sensor built inside some Nokia mobile phones (at least Nokia N95, N95 8GB, and N82). In the game you control a ball inside a labyrinth by tilting the device in your hands. Currently there are 40 different labyrinths to solve, and 3 different balls, a metal ball, a rubber ball, and a super pingpong ball. First fields are easy, but they get harder once you advance. The game requires skill, accuracy, and determination, and is highly addictive. The best time is recorded for each field, so you can also try to break <span style="position:static!important;text-decoration:underline;background-image:none!important;background-repeat:repeat!important;background-attachment:scroll!important;background-color:transparent!important;cursor:pointer!important;display:inline!important;color:#6633cc;padding-bottom:1px!important;font-size:12px;font-weight:normal;font-style:normal;font-family:Tahoma;background-position:0 50%;">the record</span> once you clear a field.</span></p>
<p style="text-align:center;"><span><!--more--><strong>Download link :</strong><a href="http://www.ziddu.com/download/5314056/marble_maze_MobliesHouse.blogspot.com.rar.html"> </a></span><a href="http://www.ziddu.com/download/5314056/marble_maze_MobliesHouse.blogspot.com.rar.html">Marble Maze</a></p>
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<title><![CDATA[ninety percent are self-inflicted.]]></title>
<link>http://whenyoureinbedyouredead.wordpress.com/2009/06/13/ninety-percent-are-self-inflicted/</link>
<pubDate>Sat, 13 Jun 2009 17:08:45 +0000</pubDate>
<dc:creator>cornolio</dc:creator>
<guid>http://whenyoureinbedyouredead.wordpress.com/2009/06/13/ninety-percent-are-self-inflicted/</guid>
<description><![CDATA[Sakka: Alright old woman, it&#8217;s my turn for you to read my future. Fortuneteller: You&#8217;ll ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><em>Sakka: Alright old woman, it&#8217;s my turn for you to read my future. </em></p>
<p><em>Fortuneteller: You&#8217;ll have countless sufferings and misfortunes.</em></p>
<p><em>Sakka: Oh yeah? How would you know, you haven&#8217;t seen my palms yet!</em></p>
<p><em>Fortuneteller: I don&#8217;t have to look at your dirty palms. The way I looked at you is enough. You&#8217;ll be mostly into troubles and pain, ninety percent of it will be self-inflicted.</em></p>
<p><em>Sakka: I told you guys it was a mistake being here with this old woman.</em></p>
<p>I got this conversation in an animated series I&#8217;ve watched before. Well, it&#8217;s just a thought that came into me when I was thinking and thinking about this certain thing boggling me.</p>
<p>She sent a group message that is a quote. Really, there&#8217;s nothing special about it. I replied with her name, an exclamation point, and a smiley trying to catch her attention and maybe start a conversation. She replied &#8220;<em>hi</em>&#8221; with my full name (She enjoys calling me that way since my first name which I didn&#8217;t use sounds cheesy) next to it and a smiley.&#8221;<em>how are you?</em>&#8221; I asked. &#8220;<em>I&#8217;m good! I&#8217;m on my way home from a date. My boyfriend and I dated.</em>&#8221; She replied. &#8220;<em>Oh ok. I&#8217;m sorry. Arrive home safely. Take care.</em>&#8221; I replied for the last time. &#8220;<em>Ok. Thank you [my first name] hahaha!</em>&#8220;</p>
<p>Everything should be fine but what I feel is not. Why on earth am I jealous with her boyfriend? I mean, I&#8217;m not in the position to do so and it&#8217;s so funny that someone got jealous of a certain subject that is in the first place, not his. Even if i could do something about it, circumstances won&#8217;t let me because the fact is, I am but a friend and not a boyfriend.</p>
<p>The reality is no one is responsible of my hurt. It&#8217;s not my friend or the guy, it is me. The pain is self-inflicted and is fanned to flame by wrong interpretations of kind friendship she&#8217;s giving me. (I admit, I&#8217;m putting much of different meanings into it LOL) I knew from the start that she had a boyfriend. That&#8217;s why I&#8217;ve marked a line of where I should place my foot on and where I should not. But when I&#8217;ve seen the two sides that the line divides, a part of me wants to jump on to the other side but my whole self established a limit and would stick to it.</p>
<p>The truth is, I entertained all this emotional chaos in me. I know things from the start and yet, I still want a tip-toe dip of how cold the river will be. We almost had a habit of texting each other every night and chat about things. We also had silly names to call each other, we shared stories and personal matters, and almost everything that bind us together. The whole thing is normal in a friendship but I love giving it other definitions and let my ears clap about it.</p>
<p>I know I am a good friend for her. I know she thought of me as a nice person and a funny guy. I know she value me because I&#8217;m not an ordinary friend she had. She cares or me, she thinks of me, she texts me, and ask if I&#8217;m doing good, because.. I am her FRIEND. Don&#8217;t you think it&#8217;s nice to have a friend that you like and give other meanings to things she does for you? LOL!! kidding!</p>
<p>Actually, I&#8217;ve broke up with my girlfriend for weeks now. (Not because of her or anyone else) I&#8217;ve realized, I need first to be matured enough before I take a <em>75 unit romantic-relationship-responsibility </em>to add to my academic subjects at school so I can manage it effectively without copromising others. So the whole thing about my friend that I am liking was just an illusion. Illusive of things that even if I had the chance to have, I still won&#8217;t and I&#8217;ve made a promise to myself to prioritize things first to my family, school, church, and to God most importantly, and least would be my love-life.</p>
<p>It is self-inflicted and It&#8217;s me that can help myself to get through this. But I am buried in a sand that felt good to me so I think I&#8217;ll gonna stay here for a while enjoying the agony. Besides there&#8217;s no one now to seriously got jealous for. Hey! I&#8217;ll be missing heartaches and all that lovey-love-love stuffs and emoticuty-chubi feelings a boyfriend-girlfriend had so let me pass on this one. LOL</p>
<p>Self-inflicted it may be, but pain in this level are manageable and easy get over with. It can be painful but I found ways on how I could enjoy it and turn it into something myself could enjoy heartbreak music and relate to it somehow. LOL!!</p>
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<title><![CDATA[Pingpong-Fan aber keinen Garten?]]></title>
<link>http://volkerracho4000.wordpress.com/2009/06/05/pingpong-fan-aber-keinen-garten/</link>
<pubDate>Fri, 05 Jun 2009 16:37:57 +0000</pubDate>
<dc:creator>isoldemaduschen</dc:creator>
<guid>http://volkerracho4000.wordpress.com/2009/06/05/pingpong-fan-aber-keinen-garten/</guid>
<description><![CDATA[Hier ist die Lösung! Besonders, wenn auch das Wohnzimmer zu klein ist: Ich finds klasse, aber gebaut]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Hier ist die Lösung! Besonders, wenn auch das Wohnzimmer zu klein ist:</p>
<p><a><img src="http://volkerracho4000.wordpress.com/files/2009/06/pingpong.jpg" alt="pingpong" title="pingpong" width="450" height="677" class="alignnone size-full wp-image-328" /></a></p>
<p>Ich finds klasse, aber gebaut ist es wohl noch nicht. <a href="http://www.yankodesign.com/2007/10/24/ping-pong-from-a-doorway/" target="_blank">Yankodesign</a> hat das ausgegraben, die Idee hatte <a href="http://tobiasfraenzel.com/" target="_blank">Tobias Fränzel</a>.</p>
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<title><![CDATA[bordtennis?]]></title>
<link>http://ordfraparken.wordpress.com/2009/05/30/bordtennis/</link>
<pubDate>Sat, 30 May 2009 12:29:53 +0000</pubDate>
<dc:creator>natten</dc:creator>
<guid>http://ordfraparken.wordpress.com/2009/05/30/bordtennis/</guid>
<description><![CDATA[]]></description>
<content:encoded><![CDATA[<div class='snap_preview'></div>]]></content:encoded>
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<title><![CDATA[Groups with free subgroups]]></title>
<link>http://lamington.wordpress.com/2009/05/28/groups-with-free-subgroups/</link>
<pubDate>Thu, 28 May 2009 21:07:40 +0000</pubDate>
<dc:creator>Danny Calegari</dc:creator>
<guid>http://lamington.wordpress.com/2009/05/28/groups-with-free-subgroups/</guid>
<description><![CDATA[More ambitious than simply showing that a group is infinite is to show that it contains an infinite ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>More ambitious than simply showing that a group is infinite is to show that it contains an infinite subgroup of a certain kind. One of the most important kinds of subgroup to study are <em>free groups</em>. Hence, one is interested in the question:</p>
<p><strong>Question</strong>: When does a group contain a (nonabelian) free subgroup?</p>
<p>Again, one can (and does) ask this question both about a specific group, and about certain classes of groups, or for a typical (in some sense) group from some given family.</p>
<p><strong>Example</strong>: If <img src='http://l.wordpress.com/latex.php?latex=%5Cmathcal%7BP%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\mathcal{P}' title='\mathcal{P}' class='latex' /> is a property of groups that is inherited by subgroups, then if no free group satisfies <img src='http://l.wordpress.com/latex.php?latex=%5Cmathcal%7BP%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\mathcal{P}' title='\mathcal{P}' class='latex' />, no group that satisfies <img src='http://l.wordpress.com/latex.php?latex=%5Cmathcal%7BP%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\mathcal{P}' title='\mathcal{P}' class='latex' /> can contain a free subgroup. An important property of this kind is <em>amenability</em>. A (discrete) group <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> is <em>amenable</em> if it admits an <em>invariant mean</em>; that is, if there is a linear map <img src='http://l.wordpress.com/latex.php?latex=m%3A+L%5E%5Cinfty%28G%29+%5Cto+%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m: L^\infty(G) \to \mathbb{R}' title='m: L^\infty(G) \to \mathbb{R}' class='latex' /> (i.e. a way to define the <em>average</em> of a bounded function over <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' />) satisfying three basic properties:</p>
<ol>
<li><img src='http://l.wordpress.com/latex.php?latex=m%28f%29+%5Cge+0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m(f) \ge 0' title='m(f) \ge 0' class='latex' /> if <img src='http://l.wordpress.com/latex.php?latex=f%5Cge+0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='f\ge 0' title='f\ge 0' class='latex' /> (i.e. the average of a non-negative function is non-negative)</li>
<li><img src='http://l.wordpress.com/latex.php?latex=m%28%5Cchi_G%29%3D1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m(\chi_G)=1' title='m(\chi_G)=1' class='latex' /> where <img src='http://l.wordpress.com/latex.php?latex=%5Cchi_G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\chi_G' title='\chi_G' class='latex' /> is the constant function taking the value <img src='http://l.wordpress.com/latex.php?latex=1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='1' title='1' class='latex' /> everywhere on <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> (i.e. the average of the constant function <img src='http://l.wordpress.com/latex.php?latex=1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='1' title='1' class='latex' /> is normalized to be <img src='http://l.wordpress.com/latex.php?latex=1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='1' title='1' class='latex' />)</li>
<li><img src='http://l.wordpress.com/latex.php?latex=m%28g%5Ccdot+f%29+%3D+m%28f%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m(g\cdot f) = m(f)' title='m(g\cdot f) = m(f)' class='latex' /> for every <img src='http://l.wordpress.com/latex.php?latex=%7B%7Dg+%5Cin+G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{}g \in G' title='{}g \in G' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=f+%5Cin+L%5E%5Cinfty%28G%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='f \in L^\infty(G)' title='f \in L^\infty(G)' class='latex' />, where <img src='http://l.wordpress.com/latex.php?latex=%28g%5Ccdot+f%29%28x%29+%3D+f%28g%5E%7B-1%7Dx%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='(g\cdot f)(x) = f(g^{-1}x)' title='(g\cdot f)(x) = f(g^{-1}x)' class='latex' /> (i.e. the mean is invariant under the obvious action of <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> on <img src='http://l.wordpress.com/latex.php?latex=L%5E%5Cinfty%28G%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='L^\infty(G)' title='L^\infty(G)' class='latex' />)</li>
</ol>
<p>If <img src='http://l.wordpress.com/latex.php?latex=H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='H' title='H' class='latex' /> is a subgroup of <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' />, there are (many) <img src='http://l.wordpress.com/latex.php?latex=H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='H' title='H' class='latex' />-invariant homomorphisms <img src='http://l.wordpress.com/latex.php?latex=j%3A+L%5E%5Cinfty%28H%29+%5Cto+L%5E%5Cinfty%28G%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='j: L^\infty(H) \to L^\infty(G)' title='j: L^\infty(H) \to L^\infty(G)' class='latex' /> taking non-negative functions to non-negative functions, and <img src='http://l.wordpress.com/latex.php?latex=%5Cchi_H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\chi_H' title='\chi_H' class='latex' /> to <img src='http://l.wordpress.com/latex.php?latex=%5Cchi_G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\chi_G' title='\chi_G' class='latex' />; for example, the (left) action of <img src='http://l.wordpress.com/latex.php?latex=H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='H' title='H' class='latex' /> on <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> breaks up into a collection of copies of <img src='http://l.wordpress.com/latex.php?latex=H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='H' title='H' class='latex' /> acting on itself, right-multiplied by a collection of right coset representatives. After choosing such a choice of representatives <img src='http://l.wordpress.com/latex.php?latex=%5Clbrace+g_%5Calpha+%5Crbrace&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\lbrace g_\alpha \rbrace' title='\lbrace g_\alpha \rbrace' class='latex' />, one for each coset <img src='http://l.wordpress.com/latex.php?latex=Hg_%5Calpha&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='Hg_\alpha' title='Hg_\alpha' class='latex' />, we can define <img src='http://l.wordpress.com/latex.php?latex=j%28f%29%28hg_%5Calpha%29+%3D+f%28h%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='j(f)(hg_\alpha) = f(h)' title='j(f)(hg_\alpha) = f(h)' class='latex' />. Composing with <img src='http://l.wordpress.com/latex.php?latex=m&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m' title='m' class='latex' /> shows that every subgroup of an amenable group is amenable (this is harder to see in the &#8220;geometric&#8221; definition of amenable groups in terms of Folner sets). On the other hand, as is well-known, a nonabelian free group is not amenable. Hence, amenable groups do not contain nonabelian free subgroups.</p>
<p>The usual way to see that a nonabelian free group is not amenable is to observe that it contains enough disjoint &#8220;copies&#8221; of big subsets. For concreteness, let <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> denote the free group on two generators <img src='http://l.wordpress.com/latex.php?latex=a%2Cb&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='a,b' title='a,b' class='latex' />, and write their inverses as <img src='http://l.wordpress.com/latex.php?latex=A%2CB&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A,B' title='A,B' class='latex' />. Let <img src='http://l.wordpress.com/latex.php?latex=W_a%2C+W_A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W_a, W_A' title='W_a, W_A' class='latex' /> denote the set of reduced words that start with either <img src='http://l.wordpress.com/latex.php?latex=a&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='a' title='a' class='latex' /> or <img src='http://l.wordpress.com/latex.php?latex=A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' />, and let <img src='http://l.wordpress.com/latex.php?latex=%5Cchi_a%2C%5Cchi_A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\chi_a,\chi_A' title='\chi_a,\chi_A' class='latex' /> denote the indicator functions of <img src='http://l.wordpress.com/latex.php?latex=W_a%2CW_A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W_a,W_A' title='W_a,W_A' class='latex' /> respectively. We suppose that <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> is amenable, and derive a contradiction. Note that <img src='http://l.wordpress.com/latex.php?latex=F+%3D+W_a+%5Ccup+aW_A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F = W_a \cup aW_A' title='F = W_a \cup aW_A' class='latex' />, so <img src='http://l.wordpress.com/latex.php?latex=m%28%5Cchi_a%29+%2B+m%28%5Cchi_A%29+%5Cge+1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m(\chi_a) + m(\chi_A) \ge 1' title='m(\chi_a) + m(\chi_A) \ge 1' class='latex' />. Let <img src='http://l.wordpress.com/latex.php?latex=V&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='V' title='V' class='latex' /> denote the set of reduced words that start with one of the strings <img src='http://l.wordpress.com/latex.php?latex=a%2CA%2Cba%2CbA&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='a,A,ba,bA' title='a,A,ba,bA' class='latex' />, and let <img src='http://l.wordpress.com/latex.php?latex=%5Cchi_V&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\chi_V' title='\chi_V' class='latex' /> denote the indicator function of <img src='http://l.wordpress.com/latex.php?latex=V&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='V' title='V' class='latex' />. Notice that <img src='http://l.wordpress.com/latex.php?latex=V&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='V' title='V' class='latex' /> is made of two disjoint copies of each of <img src='http://l.wordpress.com/latex.php?latex=W_a%2CW_A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W_a,W_A' title='W_a,W_A' class='latex' />. So on the one hand, <img src='http://l.wordpress.com/latex.php?latex=m%28%5Cchi_V%29+%5Cle+m%28%5Cchi_F%29+%3D+1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m(\chi_V) \le m(\chi_F) = 1' title='m(\chi_V) \le m(\chi_F) = 1' class='latex' />, but on the other hand, <img src='http://l.wordpress.com/latex.php?latex=m%28%5Cchi_V%29+%3D+2+%28m%28%5Cchi_a%29%2Bm%28%5Cchi_A%29%29+%5Cge+2&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m(\chi_V) = 2 (m(\chi_a)+m(\chi_A)) \ge 2' title='m(\chi_V) = 2 (m(\chi_a)+m(\chi_A)) \ge 2' class='latex' />.</p>
<p>Conversely, the usual way to show that a group <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> <em>is</em> amenable is to use the Folner condition. Suppose that <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> is finitely generated by some subset <img src='http://l.wordpress.com/latex.php?latex=S&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='S' title='S' class='latex' />, and let <img src='http://l.wordpress.com/latex.php?latex=C&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' /> denote the Cayley graph of <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> (so that <img src='http://l.wordpress.com/latex.php?latex=C&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' /> is a homogeneous locally finite graph). Suppose one can find finite subsets <img src='http://l.wordpress.com/latex.php?latex=U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U_i' title='U_i' class='latex' /> of vertices so that <img src='http://l.wordpress.com/latex.php?latex=%26%23124%3B%5Cpartial+U_i%26%23124%3B%2F%26%23124%3BU_i%26%23124%3B+%5Cto+0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='&#124;\partial U_i&#124;/&#124;U_i&#124; \to 0' title='&#124;\partial U_i&#124;/&#124;U_i&#124; \to 0' class='latex' /> (here <img src='http://l.wordpress.com/latex.php?latex=%26%23124%3BU_i%26%23124%3B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='&#124;U_i&#124;' title='&#124;U_i&#124;' class='latex' /> means the number of vertices in <img src='http://l.wordpress.com/latex.php?latex=U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U_i' title='U_i' class='latex' />, and  <img src='http://l.wordpress.com/latex.php?latex=%26%23124%3B%5Cpartial+U_i%26%23124%3B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='&#124;\partial U_i&#124;' title='&#124;\partial U_i&#124;' class='latex' /> means the number of vertices in <img src='http://l.wordpress.com/latex.php?latex=U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U_i' title='U_i' class='latex' /> that share an edge with <img src='http://l.wordpress.com/latex.php?latex=C+-+U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='C - U_i' title='C - U_i' class='latex' />). Since the &#8220;boundary&#8221; of <img src='http://l.wordpress.com/latex.php?latex=U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U_i' title='U_i' class='latex' /> is small compared to <img src='http://l.wordpress.com/latex.php?latex=U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U_i' title='U_i' class='latex' />, averaging a bounded function over <img src='http://l.wordpress.com/latex.php?latex=U_i&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U_i' title='U_i' class='latex' /> is an &#8220;almost invariant&#8221; mean; a weak limit (in the dual space to <img src='http://l.wordpress.com/latex.php?latex=L%5E%5Cinfty%28G%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='L^\infty(G)' title='L^\infty(G)' class='latex' />) is an invariant mean. Examples of amenable groups include</p>
<ol>
<li>Finite groups</li>
<li>Abelian groups</li>
<li>Unions and extensions of amenable groups</li>
<li>Groups of subexponential growth</li>
</ol>
<p>and many others. For instance, virtually solvable groups (i.e. groups containing a solvable subgroup with finite index) are amenable.</p>
<p><strong>Example</strong>: No amenable group can contain a nonabelian free subgroup. The von Neumann conjecture asked whether the converse was true. This conjecture was disproved by Olshanskii. Subsequently, Adyan <a href="http://www.ams.org/mathscinet-getitem?mr=0682486">showed</a> that the infinite free Burnside groups are not amenable. These are groups <img src='http://l.wordpress.com/latex.php?latex=B%28m%2Cn%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='B(m,n)' title='B(m,n)' class='latex' /> with <img src='http://l.wordpress.com/latex.php?latex=m%5Cge+2&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m\ge 2' title='m\ge 2' class='latex' /> generators, and subject only to the relations that the <img src='http://l.wordpress.com/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' />th power of every element is trivial. When <img src='http://l.wordpress.com/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' /> is odd and at least <img src='http://l.wordpress.com/latex.php?latex=665&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='665' title='665' class='latex' />, these groups are infinite and nonamenable. Since they are torsion groups, they do not even contain a copy of <img src='http://l.wordpress.com/latex.php?latex=%5Cmathbb%7BZ%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\mathbb{Z}' title='\mathbb{Z}' class='latex' />, let alone a nonabelian free group!</p>
<p><strong>Example</strong>: The Burnside groups are examples of groups that obey a <em>law</em>; i.e. there is a word <img src='http://l.wordpress.com/latex.php?latex=w%28x_1%2Cx_2%2C%5Ccdots%2Cx_n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w(x_1,x_2,\cdots,x_n)' title='w(x_1,x_2,\cdots,x_n)' class='latex' /> in finitely many free variables, such that <img src='http://l.wordpress.com/latex.php?latex=w%28g_1%2Cg_2%2C%5Ccdots%2Cg_n%29%3D%5Ctext%7Bid%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w(g_1,g_2,\cdots,g_n)=\text{id}' title='w(g_1,g_2,\cdots,g_n)=\text{id}' class='latex' /> for every choice of <img src='http://l.wordpress.com/latex.php?latex=g_1%2C%5Ccdots%2Cg_n+%5Cin+G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g_1,\cdots,g_n \in G' title='g_1,\cdots,g_n \in G' class='latex' />. For example, an abelian group satisfies the law <img src='http://l.wordpress.com/latex.php?latex=x_1x_2x_1%5E%7B-1%7Dx_2%5E%7B-1%7D%3D%5Ctext%7Bid%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x_1x_2x_1^{-1}x_2^{-1}=\text{id}' title='x_1x_2x_1^{-1}x_2^{-1}=\text{id}' class='latex' />. Evidently, a group that obeys a law does not contain a nonabelian free subgroup. However, there are examples of groups which do not obey a law, but which also do not contain any nonabelian free subgroup. An example is the classical <em>Thompson&#8217;s group</em> <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' />, which is the group of orientation-preserving piecewise-linear homeomorphisms of <img src='http://l.wordpress.com/latex.php?latex=%5B0%2C1%5D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='[0,1]' title='[0,1]' class='latex' /> with finitely many breakpoints at dyadic rationals (i.e. points of the form <img src='http://l.wordpress.com/latex.php?latex=p%2F2%5Eq&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p/2^q' title='p/2^q' class='latex' /> for integers <img src='http://l.wordpress.com/latex.php?latex=p%2Cq&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p,q' title='p,q' class='latex' />) and with slopes integral powers of <img src='http://l.wordpress.com/latex.php?latex=2&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='2' title='2' class='latex' />. To see that this group does not obey a law, one can show (quite easily) that in fact <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> is dense (in the <img src='http://l.wordpress.com/latex.php?latex=C%5E0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='C^0' title='C^0' class='latex' /> topology) in the group <img src='http://l.wordpress.com/latex.php?latex=%5Ctext%7BHomeo%7D%5E%2B%28%5B0%2C1%5D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\text{Homeo}^+([0,1])' title='\text{Homeo}^+([0,1])' class='latex' /> of <em>all</em> orientation-preserving homeomorphisms of the interval. This latter group contains nonabelian free groups; by approximating the generators of such a group arbitrarily closely, one obtains pairs of elements in <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> that do not satisfy any identity of length shorter than any given constant. On the other hand, a famous theorem of <a href="http://www.ams.org/mathscinet-getitem?mr=0782231">Brin-Squier</a> says that <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> does not contain any nonabelian free subgroup. In fact, the entire group <img src='http://l.wordpress.com/latex.php?latex=%5Ctext%7BPL%7D%5E%2B%28%5B0%2C1%5D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\text{PL}^+([0,1])' title='\text{PL}^+([0,1])' class='latex' /> does not contain any nonabelian free subgroup. A short proof of this fact can be found in <a href="http://arxiv.org/abs/math/0607482">my paper</a> as a corollary of the fact that every subgroup <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> of <img src='http://l.wordpress.com/latex.php?latex=%5Ctext%7BPL%7D%5E%2B%28%5B0%2C1%5D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\text{PL}^+([0,1])' title='\text{PL}^+([0,1])' class='latex' /> has vanishing stable commutator length; since stable commutator length is nonvanishing in nonabelian free groups, this shows that there are no such subgroups of <img src='http://l.wordpress.com/latex.php?latex=%5Ctext%7BPL%7D%5E%2B%28%5B0%2C1%5D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\text{PL}^+([0,1])' title='\text{PL}^+([0,1])' class='latex' />. (Incidentally, and complementarily, there is a very short proof that stable commutator length vanishes on any group that obeys a law; we will give this proof in a subsequent post).</p>
<p><strong>Example</strong>: If <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> surjects onto <img src='http://l.wordpress.com/latex.php?latex=H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='H' title='H' class='latex' />, and <img src='http://l.wordpress.com/latex.php?latex=H&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='H' title='H' class='latex' /> contains a free subgroup <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' />, then there is a section from <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> to <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> (by freeness), and therefore <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> contains a free subgroup.</p>
<p><strong>Example</strong>: The most useful way to show that <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> contains a nonabelian free subgroup is to find a suitable action of <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> on some space <img src='http://l.wordpress.com/latex.php?latex=X&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='X' title='X' class='latex' />. The following is known as Klein&#8217;s ping-pong lemma. Suppose one can find disjoint subsets <img src='http://l.wordpress.com/latex.php?latex=U%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U^\pm' title='U^\pm' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=V%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='V^\pm' title='V^\pm' class='latex' /> of <img src='http://l.wordpress.com/latex.php?latex=X&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='X' title='X' class='latex' />, and elements <img src='http://l.wordpress.com/latex.php?latex=g%2Ch+%5Cin+G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g,h \in G' title='g,h \in G' class='latex' /> so that <img src='http://l.wordpress.com/latex.php?latex=g%28U%5E%2B+%5Ccup+V%5E%5Cpm%29+%5Csubset+U%5E%2B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g(U^+ \cup V^\pm) \subset U^+' title='g(U^+ \cup V^\pm) \subset U^+' class='latex' />, <img src='http://l.wordpress.com/latex.php?latex=g%5E%7B-1%7D%28U%5E-+%5Ccup+V%5E%5Cpm%29+%5Csubset+U%5E-&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g^{-1}(U^- \cup V^\pm) \subset U^-' title='g^{-1}(U^- \cup V^\pm) \subset U^-' class='latex' />, and similarly interchanging the roles of <img src='http://l.wordpress.com/latex.php?latex=U%5E%5Cpm%2C+V%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U^\pm, V^\pm' title='U^\pm, V^\pm' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=g%2Ch&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g,h' title='g,h' class='latex' />. If <img src='http://l.wordpress.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w' title='w' class='latex' /> is a reduced word in <img src='http://l.wordpress.com/latex.php?latex=g%5E%7B%5Cpm+1%7D%2Ch%5E%7B%5Cpm+1%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g^{\pm 1},h^{\pm 1}' title='g^{\pm 1},h^{\pm 1}' class='latex' />, one can follow the trajectory of a point under the orbit of subwords of <img src='http://l.wordpress.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w' title='w' class='latex' /> to verify that <img src='http://l.wordpress.com/latex.php?latex=w&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w' title='w' class='latex' /> is nontrivial. The most common way to apply this in practice is when <img src='http://l.wordpress.com/latex.php?latex=g%2Ch&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g,h' title='g,h' class='latex' /> act on <img src='http://l.wordpress.com/latex.php?latex=X&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='X' title='X' class='latex' /> with <em>source-sink dynamics</em>; i.e. the element <img src='http://l.wordpress.com/latex.php?latex=g&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g' title='g' class='latex' /> has two fixed points <img src='http://l.wordpress.com/latex.php?latex=u%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='u^\pm' title='u^\pm' class='latex' /> so that every other point converges to <img src='http://l.wordpress.com/latex.php?latex=u%5E%2B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='u^+' title='u^+' class='latex' /> under positive powers of <img src='http://l.wordpress.com/latex.php?latex=g&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g' title='g' class='latex' />, and to <img src='http://l.wordpress.com/latex.php?latex=u%5E-&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='u^-' title='u^-' class='latex' /> under negative powers of <img src='http://l.wordpress.com/latex.php?latex=g&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g' title='g' class='latex' />. Similarly, <img src='http://l.wordpress.com/latex.php?latex=h&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='h' title='h' class='latex' /> has two fixed points <img src='http://l.wordpress.com/latex.php?latex=v%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='v^\pm' title='v^\pm' class='latex' /> with similar dynamics. If the points <img src='http://l.wordpress.com/latex.php?latex=u%5E%5Cpm%2Cv%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='u^\pm,v^\pm' title='u^\pm,v^\pm' class='latex' /> are disjoint, and <img src='http://l.wordpress.com/latex.php?latex=X&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='X' title='X' class='latex' /> is compact, one can take any small open neighborhoods <img src='http://l.wordpress.com/latex.php?latex=U%5E%5Cpm%2CV%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='U^\pm,V^\pm' title='U^\pm,V^\pm' class='latex' /> of <img src='http://l.wordpress.com/latex.php?latex=u%5E%5Cpm%2Cv%5E%5Cpm&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='u^\pm,v^\pm' title='u^\pm,v^\pm' class='latex' />, and then sufficiently large powers of <img src='http://l.wordpress.com/latex.php?latex=g&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g' title='g' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=h&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='h' title='h' class='latex' /> will satisfy the hypotheses of ping-pong.</p>
<p><strong>Example</strong>: Every hyperbolic group <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> acts on its Gromov boundary <img src='http://l.wordpress.com/latex.php?latex=%5Cpartial_%5Cinfty+G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\partial_\infty G' title='\partial_\infty G' class='latex' />. This boundary is the set of equivalence classes of quasigeodesic rays in (the Cayley graph of) <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' />, where two rays are equivalent if they are a finite Hausdorff distance apart. Non-torsion elements act on the boundary with source-sink dynamics. Consequently, every pair of non-torsion elements in a hyperbolic group either generate a virtually cyclic group, or have powers that generate a nonabelian free group.</p>
<p>It is striking to see how easy it is to construct nonabelian free subgroups of a hyperbolic group, and how difficult to construct closed surface subgroups. We will return to the example of hyperbolic groups in a future post.</p>
<p><strong>Example</strong>: The <em>Tits alternative</em> says that any linear group <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> (i.e. any subgroup of <img src='http://l.wordpress.com/latex.php?latex=%5Ctext%7BGL%7D%28n%2C%5Cmathbb%7BR%7D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\text{GL}(n,\mathbb{R})' title='\text{GL}(n,\mathbb{R})' class='latex' /> for some <img src='http://l.wordpress.com/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' />) either contains a nonabelian free subgroup, or is virtually solvable (and therefore amenable). This can be derived from ping-pong, where <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> is made to act on certain spaces derived from the linear action (e.g. locally symmetric spaces compactified in certain ways, and buildings associated to discrete valuations on the ring of entries of matrix elements of <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' />). </p>
<p><strong>Example</strong>: There is a Tits alternative for subgroups of other kinds of groups, for example mapping class groups, as shown by <a href="http://www.ams.org/mathscinet-getitem?mr=0745513">Ivanov</a> and <a href="http://www.ams.org/mathscinet-getitem?mr=0800253">McCarthy</a>. The mapping class group (of a surface) acts on the Thurston boundary of Teichmuller space. Every subgroup of the mapping class group either contains a nonabelian free subgroup, or is virtually abelian. Roughly speaking, either elements move points in the boundary with enough dynamics to be able to do ping-pong, or else the action is &#8220;localized&#8221; in a train-track chart, and one obtains a linear representation of the group (enough to apply the ordinary Tits alternative). Virtually solvable subgroups of mapping class groups are virtually abelian.</p>
<p><strong>Example</strong>: A similar Tits alternative holds for <img src='http://l.wordpress.com/latex.php?latex=%5Ctext%7BOut%7D%28F_n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\text{Out}(F_n)' title='\text{Out}(F_n)' class='latex' />. This was shown by Bestvina-Feighn-Handel in <a href="http://arxiv.org/abs/math/9712217">these</a> <a href="http://arxiv.org/abs/math/9712218">three</a> <a href="http://arxiv.org/abs/math/9712219">papers</a> (the third paper shows that solvable subgroups are virtually abelian, thus emphasizing the parallels with mapping class groups).</p>
<p><strong>Example</strong>: If <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> is a finitely generated group of homeomorphisms of <img src='http://l.wordpress.com/latex.php?latex=S%5E1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='S^1' title='S^1' class='latex' />, then there is a kind of Tits alternative, first proposed by Ghys, and proved by <a href="http://www.ams.org/mathscinet-getitem?mr=1797749">Margulis</a>: either <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> preserves a probability measure on <img src='http://l.wordpress.com/latex.php?latex=S%5E1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='S^1' title='S^1' class='latex' /> (which might be singular), or it contains a nonabelian free subgroup. To see this, first note that either <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> has a finite orbit (which supports an invariant probability measure) or the action is semi-conjugate to a minimal action (one with all orbits dense). In the second case, the proof depends on understanding the centralizer of the group action: either the centralizer is infinite, in which case the group is conjugate to a group of rotations, or it is finite cyclic, and one obtains an action of <img src='http://l.wordpress.com/latex.php?latex=G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G' title='G' class='latex' /> on a &#8220;smaller&#8221; circle, by quotienting out by the centralizer. So one may assume the action is minimal with trivial centralizer. In this case, one shows that the action has the property that for any nonempty intervals <img src='http://l.wordpress.com/latex.php?latex=I%2CJ&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='I,J' title='I,J' class='latex' /> in <img src='http://l.wordpress.com/latex.php?latex=S%5E1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='S^1' title='S^1' class='latex' />, there is some <img src='http://l.wordpress.com/latex.php?latex=%7B%7Dg+%5Cin+G&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{}g \in G' title='{}g \in G' class='latex' /> with <img src='http://l.wordpress.com/latex.php?latex=g%28I%29+%5Csubset+J&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g(I) \subset J' title='g(I) \subset J' class='latex' />; i.e. any interval may be put inside any other interval by some element of the group. For such an action, it is very easy to do ping-pong. Incidentally, a minor variation on this result, and with essentially this argument, was established by <a href="http://arxiv.org/abs/math/9712268">Thurston</a> in the context of uniform foliations of <img src='http://l.wordpress.com/latex.php?latex=3&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='3' title='3' class='latex' />-manifolds before Ghys proposed his question.</p>
<p><strong>Example</strong>: If <img src='http://l.wordpress.com/latex.php?latex=%5Crho_t&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\rho_t' title='\rho_t' class='latex' /> is an (algebraic) family of representations of a (countable) free group <img src='http://l.wordpress.com/latex.php?latex=F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='F' title='F' class='latex' /> into an algebraic group, then either some element <img src='http://l.wordpress.com/latex.php?latex=g+%5Cin+F&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='g \in F' title='g \in F' class='latex' /> is in the kernel of every <img src='http://l.wordpress.com/latex.php?latex=%5Crho_t&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\rho_t' title='\rho_t' class='latex' />, or the set of faithful representations is &#8220;generic&#8221;, i.e. the intersection of countably many open dense sets. This is because the set of representations for which a given element is in the kernel is Zariski closed, and therefore its complement is open and either empty or dense (one must add suitable hypotheses or conditions to the above to make it rigorous).</p>
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<title><![CDATA[Di Marunong Mag-Pingpong]]></title>
<link>http://2009summer.wordpress.com/2009/05/13/di-marunong-mag-pingpong/</link>
<pubDate>Tue, 12 May 2009 21:23:07 +0000</pubDate>
<dc:creator>Ser Joshua Lagrimas</dc:creator>
<guid>http://2009summer.wordpress.com/2009/05/13/di-marunong-mag-pingpong/</guid>
<description><![CDATA[Ser Joshua Lagrimas [Week4 Part2] Grabe,  parang may hangover pa rin ako sa naganap na Code Review n]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Ser <strong>Josh</strong>ua Lagrimas <em>[Week4 Part2]</em></p>
<p>Grabe,  parang may hangover pa rin ako sa naganap na Code Review nung isang araw. Nahigop niya talaga lahat ng lakas ko. Pero ang katapusan ng Code Review na ito ay isa ring senyales na simula na naman kami sa regular na pagtuturo sa Java Boot Camp. Patagal ng patagal napapansin kong mas komportable na kami sa isa&#8217;t isa. Kahit yung mga nagtuturo sa min, kahit paiba-iba, ewan ko ba, mas wala ng tensyon di gaya nung mga nakaraang araw.</p>
<p><!--more-->Ngayon naman at nagsimula na kami sa Ikalawang Module, tinuturuan naman kami ng Web Development. Sa kinasamaang-palad, hindi kasama sa kurikulum ng Departamento ng Computer Science ni isang Web Development na asignatura. Ito ako ngayon takot na takot gumawa lang ng simpleng &#8220;tag&#8221;. May payak na kaalaman ako sa HTML pero hindi siya sapat para bigyan ako ng lakas ng loob para gumamit at gumawa ng isang simpleng pahina sa Web.</p>
<p>Habang naglalaro ang mga tao sa labas ng Pingpong, sayang-saya naman kaming mga Dev trainees sa pakikinig ng turo, kung nakikinig nga. (Kayo na lang ang humusga.)</p>
<div id="attachment_2466" class="wp-caption aligncenter" style="width: 430px"><img class="size-full wp-image-2466" title="Dev Trainees Inside The Technical Excellence Room" src="http://2009summer.wordpress.com/files/2009/05/3225262_1e48f1b3e2767aa4c5db4b2b6625d79e.jpg" alt="habang &#34;nakikinig&#34; sa lecture sa harap.." width="420" height="315" /><p class="wp-caption-text">habang &#34;nakikinig&#34; sa lecture sa harap..</p></div>
<p>Ang saya noh?XD Ganyan kami lagi sa loob ng training room. Araw-araw, oras-oras at minu-minuto.</p>
<p>Ayun nga, kaya nga ako nag-blog kasi ibabahagi ko sana ang isa sa mga tinuro sa min mula sa Ikalawang Module, ang <a href="http://en.wikipedia.org/wiki/JavaServer_Pages">JSP (JavaServer Pages)</a> at <a href="http://en.wikipedia.org/wiki/JSTL">JSTL (JavaServer Pages Standard Tag Library)</a>. Ilan ito sa mga teknolohiyang pwede nating magamit sa paggawa ng &#8220;V&#8221; sa M-V-C, ang &#8220;View&#8221;.</p>
<p>Ang JSP kasi ay isang teknolohiya sa Java na binibigyang kapangyarihan ang mga developers na lagyan ng code ng Java yung mismong web page file. Ito rin ay nagdadagdag ng tag libraries na nagsisilbing palugit ng pangunahing <strong><span style="font-weight:normal;"><a href="http://en.wikipedia.org/wiki/XML">XML</a></span></strong><a href="http://en.wikipedia.org/wiki/XML"> (</a><strong><span style="font-weight:normal;"><a href="http://en.wikipedia.org/wiki/XML">Extensible Markup Language</a></span></strong><a href="http://en.wikipedia.org/wiki/XML">)</a> at <a href="http://en.wikipedia.org/wiki/Html">HTML (HyperText Markup Language)</a> tags.</p>
<p>Sa JSP, tinuruan kami ng:</p>
<ul>
<li><a href="http://72.5.124.55/javaee/5/docs/tutorial/doc/bnaos.html">Declarations</a></li>
<li><a href="http://72.5.124.55/javaee/5/docs/tutorial/doc/bnaov.html">Expressions</a></li>
<li><a href="http://72.5.124.55/javaee/5/docs/tutorial/doc/bnaou.html">Scriptlets</a></li>
<li><a href="http://www.codemiles.com/jsp-examples/jsp-comments-t3358.html">Comments</a></li>
<li><a href="http://nlc.nlc.go.cn/resin-doc/jsp/directives.xtp">Directives</a></li>
<li><a href="http://nlc.nlc.go.cn/resin-doc/jsp/actions.xtp">Actions</a></li>
<li><a href="http://www.exforsys.com/tutorials/jsp/jsp-implicit-and-session-objects.html">Implicit JSP Objects</a></li>
<li><a href="http://java.sun.com/developer/EJTechTips/2003/tt0114.html">Error Pages</a></li>
<li><a href="http://www.stardeveloper.com/articles/display.html?article=2001072001&#38;page=1">JavaBean</a></li>
</ul>
<p>Nakakatawa kasi habang nasa loob kami bawat pagkakataon na lumalabas ako para mag-CR, nakikita kong naglalaro ng Pingpong yung mga tao sa labas. At ang nakakagulat pa nito, mismo sina Sir Joel at Sir Butch pa yung naglalaro.</p>
<div id="attachment_2472" class="wp-caption aligncenter" style="width: 430px"><img class="size-full wp-image-2472" title="Bosses Playing Pingpong" src="http://2009summer.wordpress.com/files/2009/05/3225262_9b19e78dac960f891ca612e108fc813b.jpg" alt="Sir Joel at Sir Butch naglalaban.." width="420" height="315" /><p class="wp-caption-text">Sir Joel at Sir Butch naglalaban..</p></div>
<p>Ayun kaya pagbalik ko mula sa CR, tinuturuan na kaming mag-JSTL. Ano nga ba ang isang <a href="http://www.jsptut.com/Tags.jsp">JSP Tag</a>? Katulad lang siya ng isang HTML Tag pero dito may mga dagdag na functionalities na hindi kayang gawin ng isang regular na tag.  Halimbawa na lang ng mga kayang gawin sa JSP Tag ay mga simpleng pagse-set at pagtatanggal ng variables, paggamit ng <a href="http://java.sun.com/j2ee/1.4/docs/tutorial/doc/JSPIntro7.html">EL (Expression Language)</a>, Conditionals (if-else), Iterators (forEach), etc. Marami pa siyempre pero sa tingin ko sapat na yan para mailarawan anung meron sa JSTL.</p>
<p>Actually, pwede rin kaming maglaro sana ng Pingpong nung araw na yun pero siyempre di pwede kasi may lecture. </p>
<div id="attachment_2473" class="wp-caption aligncenter" style="width: 430px"><img class="size-full wp-image-2473" title="Employees vs Interns" src="http://2009summer.wordpress.com/files/2009/05/3225262_fdd9f86c88ef4ad51a0c3f4e3f4d391f.jpg" alt="si Thea representative ng interns.. go Thea!" width="420" height="315" /><p class="wp-caption-text">si Thea representative ng interns.. go Thea!</p></div>
<p>Nakakainggit nga PMBA interns eh kasi laro lang sila ng laro habang kaming mga Dev busyng-busy. (Hindi naman halatang bitter noh?)</p>
<p> </p>
<p> </p>
<p>PS</p>
<p>Nang sumunod na araw, nasubukan kong maglaro ng Pingpong. Ayun, natuklasan ko lang kung gaano ako ka-bobo maglaro. Grabe, walang kontrol sa pagpalo at walang direksyon. Hay.</p>
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<title><![CDATA[La Bellevilloise]]></title>
<link>http://soixantehuit.wordpress.com/2009/05/12/la-bellevilloise/</link>
<pubDate>Tue, 12 May 2009 17:44:33 +0000</pubDate>
<dc:creator>soixantehuit</dc:creator>
<guid>http://soixantehuit.wordpress.com/2009/05/12/la-bellevilloise/</guid>
<description><![CDATA[I&#8217;d been meaning to go to La Bellevilloise for a few weeks and finally took the chance to visi]]></description>
<content:encoded><![CDATA[I&#8217;d been meaning to go to La Bellevilloise for a few weeks and finally took the chance to visi]]></content:encoded>
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<title><![CDATA[Metiendo pelotitas ]]></title>
<link>http://enlacesweb.wordpress.com/2009/05/10/metiendo-pelotitas/</link>
<pubDate>Sun, 10 May 2009 08:55:14 +0000</pubDate>
<dc:creator>n0re1</dc:creator>
<guid>http://enlacesweb.wordpress.com/2009/05/10/metiendo-pelotitas/</guid>
<description><![CDATA[Mira lo que hacen estos chicos con pelotas de ping pong.]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img class="aligncenter" title="pelotas pingpong" src="http://mibrujula.com/imagenes_noticias/2009/ping.jpg" alt="" width="390" height="150" /></p>
<h6>Mira lo que hacen estos chicos <a href="http://www.mibrujula.com/videos/Metiendo-pelotitas_V495.html" target="_blank"><span style="color:#ff6600;">con pelotas de ping pong</span></a>.</h6>
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<title><![CDATA[Tenis stołowy - topspin (forehand)]]></title>
<link>http://bibliotekaswiatow.wordpress.com/2009/05/05/tenis-stolowy-topspin-forehand/</link>
<pubDate>Tue, 05 May 2009 14:27:16 +0000</pubDate>
<dc:creator>chmuriat</dc:creator>
<guid>http://bibliotekaswiatow.wordpress.com/2009/05/05/tenis-stolowy-topspin-forehand/</guid>
<description><![CDATA[Topspin z forhendu jest poza zbiciem najbardziej dynamicznym i silnym zagraniem w tenisie stołowym. ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Topspin z forhendu jest poza zbiciem najbardziej dynamicznym i silnym zagraniem w tenisie stołowym. Kiedy jest wykonywany poprawnie może być podstawą Twojej gry. Kluczem do nauki topspinu z forhendu jest porównanie go do podobnych zagrań w innych sportach. Może to brzmieć zabawnie dla profesjonalnego gracza, ale takie porównania mają swoje uzasadnienie.</p>
<p>Jeśli o topspinie będziesz myślał tylko w kategoriach zagrania pingpongowego to może ci się wydawać, że podstawą tego zagrania jest ruch ręką. Nic bardziej mylnego. Poprawny topspin wymaga zaangażowania całego ciała.  Pod tym względem jest on podobny do forhendu w tenisie ziemnym, uderzenia bejsbolowego czy golfowego.</p>
<p>Wyobraź sobie bejsbolistów (np. Marka McGuire’a czy Ken’a Griffeya – to ci bardziej znani <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_smile.gif' alt=':)' class='wp-smiley' /> . Czy mogliby tak daleko uderzać piłkę gdyby nie wprawiali w ruch całego swojego ciała, poczynając od nóg, tułowia i mięśni brzucha? Wyobraź sobie perfekcyjne zagranie Tigera Woodsa. Czy byłoby ono możliwe bez przenoszenia ciężaru z jednej nogi na drugą? Wyobraź sobie forhend Andre Agassiego. Czy byłoby to takie wspaniałe zagranie gdyby nie podchodził nisko pod piłkę? Na wszystkie pytania odpowiedź brzmi „nie”. Powinno dać ci to do myślenia kiedy będziesz uczył się tego zagrania.</p>
<p>Najbardziej istotne w topspinie jest energiczne przeniesienie ciężaru ciała z prawej nogi na lewą (dla graczy leworęcznych – odwrotnie). Jest to równorzędne z decyzją o nadaniu piłeczce większej szybkości lub rotacji. Jeśli zależy ci na większej rotacji ruch ręką musi iść bardziej w górę. Kontakt rakietki z piłeczką musi być możliwie krótki. Aby uzyskać więcej szybkości przenosisz ciężar ciała bardziej w górę. Kontakt piłeczki z rakietką jest dłuższy.</p>
<p>Topspin na zagrania podcięte (z rotacją wsteczną)</p>
<p>Aby zagrać topspin przeciw zagraniu podciętemu musisz zrobić krok w kierunku piłki, następnie przyjąć odpowiednią pozycję (patrz foto), zrobić skręt tułowia i następnie energicznie wykonać uderzenie przenosząc ciężar ciała na lewą nogę. Podczas zagrania prawa noga powinna być lekko wysunięta do tyłu, a ciało skręcone ok. 45 st. w stosunku do blatu stołu. Jeśli masz więcej czasu na zagranie możesz zrobić większe wychylenie. W dzisiejszym tenisie stołowym zawodnicy nie mają już tyle czasu na pełne zagranie ciałem, jednak nie powinno się o tym zapominać, gdyż sam ruch ręką może wypaczyć całe uderzenie. Pamiętaj, że przy zagraniach powinieneś maksymalnie rozluźnić ramię i rękę.</p>
<p>Podczas zagrania topspinu tułów i ramię wykonuje obszerniejszy skręt niż nogi. Piłkę należy zagrywać kiedy jest w najwyższym punkcie lub kiedy jest opadająca, w zależności od tego czy chcemy uzyskać większą rotację lub szybkość uderzenia. Zagranie powinno kończyć się na wysokości głowy lub w górnej części klatki piersiowej. Ważna jest również pozycja głowy. W początkowej fazie powinna być pochylona w stosunku do stołu, a w momencie kończenia uderzenia powinna być wychylona do góry. Ważne jest aby mieć kontakt wzrokowy z piłeczką. Przed i po zagraniu głowa powinna podążać za torem lotu piłeczki.</p>
<p>Kiedy już opanujesz grę topspinem z forhendu powinieneś skupić się na urozmaicaniu swojej rotacji. Z czasem przekonasz się, że wcale nie takie trudne jest zagrywanie topspinów z rotacją boczną (sidespinów) czy tzw. fast-spinów (silnych topsinów pozbawionych niemal rotacji). To jest jednak temat na inny artykuł&#8230;</p>
<p>Opracowanie: blu, Sean Lonergan</p>
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