<?xml version="1.0" encoding="UTF-8"?><!-- generator="wordpress.com" -->
<rss version="2.0"
	xmlns:content="http://purl.org/rss/1.0/modules/content/"
	xmlns:wfw="http://wellformedweb.org/CommentAPI/"
	xmlns:dc="http://purl.org/dc/elements/1.1/"
	>

<channel>
	<title>inversions &amp;laquo; WordPress.com Tag Feed</title>
	<link>http://en.wordpress.com/tag/inversions/</link>
	<description>Feed of posts on WordPress.com tagged "inversions"</description>
	<pubDate>Sat, 02 Jan 2010 17:01:21 +0000</pubDate>

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

<item>
<title><![CDATA[Easy Inversion for Beauty (Instant Facelift!), Calmness and Peace ]]></title>
<link>http://kimberlysnyder.net/2009/12/15/easy-inversion-for-beauty-instant-facelift-calmness-and-peace/</link>
<pubDate>Tue, 15 Dec 2009 22:01:13 +0000</pubDate>
<dc:creator>kimberlysnyder</dc:creator>
<guid>http://kimberlysnyder.net/2009/12/15/easy-inversion-for-beauty-instant-facelift-calmness-and-peace/</guid>
<description><![CDATA[Hey Guys! Inversions are great for mental clarity, stimulating the endocrine system, bringing a beau]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><span style="font-family:Arial,Helvetica,sans-serif;font-size:x-small;"><a rel="nofollow" href="http://www.youtube.com/watch?v=BZhp-jby3NQ" target="_blank"><span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/BZhp-jby3NQ&#038;rel=1&#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/BZhp-jby3NQ&#038;rel=1&#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><br />
</a></span></p>
<p>Hey Guys! Inversions are great for mental clarity, stimulating the endocrine system, bringing a beautiful glow to the face, and bringing prana to our upper chakras. The fresh blood flow and oxygen is indeed like getting an instant facelift!</p>
<p>Hare pose (not Hair pose! <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_smile.gif' alt=':)' class='wp-smiley' />  ), is an easy inversion almost everyone can do. If you are more flexible, you can try holding on to your ankles the whole time, and bringing your head down closer to your knees.</p>
<p>Namaste and Shanti!</p>
<p>Kimberly</p>
<p><span style="font-family:Arial,Helvetica,sans-serif;font-size:x-small;"><br />
</span></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Off the wall: the yoga class that will get your feet in the air]]></title>
<link>http://wellandgoodnyc.wordpress.com/2009/11/30/off-the-wall-the-yoga-class-that-will-get-your-feet-in-the-air/</link>
<pubDate>Mon, 30 Nov 2009 13:00:28 +0000</pubDate>
<dc:creator>Well+Good</dc:creator>
<guid>http://wellandgoodnyc.wordpress.com/2009/11/30/off-the-wall-the-yoga-class-that-will-get-your-feet-in-the-air/</guid>
<description><![CDATA[Many yoga practitioners find they hit a wall when it comes to inversions, literally and figuratively]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img class="alignleft size-medium wp-image-2435" title="Strong Inversion" src="http://wellandgoodnyc.wordpress.com/files/2009/11/inversions1.jpg?w=300" alt="" width="290" height="225" />Many yoga practitioners find they hit a wall when it comes to inversions, literally and figuratively. Standing poses, arm balances, tricky transitions all seem to be coming along, but balancing inversions beyond a basic headstand in the center of the room seem next to impossible.</p>
<p>This leads us to one of the great yoga conundrums: Practicing handstand with a wall does very little to teach you how to practice handstand without one.</p>
<p>To learn how to take your handstand to the center of the room—and into the middle of your vinyasa—a <a href="http://www.multi-intenso.com" target="_blank">Multi-Intenso </a>class with David Regelin may be the answer.</p>
<p>“The first step is really deciding you want to do a handstand,” says Regelin, founder of Multi-Intenso, a vinyasa style that draws upon classical poses, his own creations, and a &#8220;serious emphasis on inversions.&#8221; Taught at <a href="http://www.kulayoga.com">Kula Yoga Project</a>, <a href="http://www.theshala.com">The Shala</a>, and <a href="http://www.pureyoga.com">Pure Yoga</a>, Regelin’s class gets you past halfhearted hopping into handstand by introducing a choreographed sequence that combines integrated core work, conditioning, and openings (hamstrings, look out). Multi-Intenso creates the confidence as well as the balance, strength, and flexibility needed to get into, stay in, and get out of handstand gracefully. (Our own progress is proof.)</p>
<div id="attachment_2436" class="wp-caption alignright" style="width: 310px"><img class="size-medium wp-image-2436" title="Strong Inversion 2" src="http://wellandgoodnyc.wordpress.com/files/2009/11/inversions2.jpg?w=300" alt="" width="300" height="225" /><p class="wp-caption-text">Cristina Tarantola has been taking David Regelin&#39;s classes for almost two years, 3 times a week</p></div>
<p>Still worried about collateral damage? Regelin recommends you have an exit plan—a way to get down safely that includes yourself and those around you, should you lose your balance. (The one-foot-down-at-a-time plan is a good option.) And give it some time: Even with Regelin&#8217;s classes, really nailing inversions might demand more patience from your practice than strategy.</p>
<p><em>Look for Multi-Intenso at Kula Yoga Project, 28 Warren St., nr. Church, City Hall, 212-945-4460, <a href="http://www.kulayoga.com" target="_blank">www.kulayoga.com</a>, $17 single class; The Shala, 816 Broadway, at 12th St., Union Square, 212-979-9988, <a href="http://www.theshala.com" target="_blank">www.theshala.com</a>, $18 single class; Pure Yoga, 203 East 86th St., at Third Ave., Upper East Side, 212-360-1888, <a href="http://www.pureyoga.com" target="_blank">www.pureyoga.com</a>, $30 day pass</em><br />
<a class="addthis_button" href="http://www.addthis.com/bookmark.php?v=250&#38;pub=wellandgoodnyc"><img src="http://s7.addthis.com/static/btn/lg-share-en.gif" alt="Bookmark and Share" width="125" height="16" /></a><br />
<!-- AddThis Button END --><em><br />
How long have you been working on your free-standing inversions? Got any tips? Tell us, here!</em></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[When Do I Use Chord Inversions?]]></title>
<link>http://garyewer.wordpress.com/2009/11/24/when-do-i-use-chord-inversions/</link>
<pubDate>Tue, 24 Nov 2009 17:48:41 +0000</pubDate>
<dc:creator>garyewer</dc:creator>
<guid>http://garyewer.wordpress.com/2009/11/24/when-do-i-use-chord-inversions/</guid>
<description><![CDATA[Written by Gary Ewer, from “The Essential Secrets of Songwriting” website. Follow Gary on Twitter fo]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Written by Gary Ewer, from “<a href="http://www.secretsofsongwriting.com">The Essential Secrets of Songwriting</a>” website.<br />
<a href="http://www.twitter.com/garyewer" target="_blank"><strong>Follow Gary on Twitter</strong></a> for songwriting tips, news and more.<br />
__________________</p>
<p><img class="alignleft size-full wp-image-990" style="margin-left:5px;margin-right:5px;" title="Music Paper" src="http://garyewer.wordpress.com/files/2009/11/photo46.jpg" alt="" width="120" height="80" />An inverted chord means that you&#8217;ve moved the root of a chord to some upper position, leaving a note other than the root as the lowest sounding note. It&#8217;s a really great device that will add colour to your musical palette. But chord inversions have to be treated with a bit of care; randomly opting for an inversion can lead to a chord progression that sounds a bit disorganized.</p>
<p>A triad is a three-note chord that includes a root, a third and a fifth. For example, in C major, a C triad is comprised of a C, E and G played simultaneously. With the root on the bottom, you get this triad in its most stable position.</p>
<p>By moving the root upward and away from its bottom position, you now have the 3rd of the chord as the lowest-sounding note (C/E). This triad, called a first inversion triad, is slightly less stable, in the sense that we usually don&#8217;t end musical phrases with this type of chord. Ends of phrases need the solidity that comes from having the root at the bottom.</p>
<p>One other possibility for triads is to place the 5th at the bottom, and this gives us an even-less stable triad (C/G). This chord begs for a resolution to G (i.e., C/G  - G &#8211; C).</p>
<p>But to say that a triad is unstable is by no means a criticism. In fact, you will find that musical progressions sound fresh and interesting when they are comprised of a balance of root position and inverted chords. The question is&#8230; how do we know when to use them?</p>
<p>Here are some quick tips to guide you:</p>
<ol>
<li><strong>Use an inversion to animate a static bass line.</strong> If your progression requires your bass to do a lot of jumping by 4ths and 5ths, you can smooth this out and get the line moving by using an inversion. Here&#8217;s an example: C  F  C  G  C, with each chord played for 4 beats. Try modifying it with inversions (each chord played for 2 beats:) C  C/E  F  F/A  C  C  G  G/B  C</li>
<li><strong>Use an inversion to make a boring progression more interesting.</strong> Related to point 1 above, if your progression has you dwelling on one chord for a long period of time, get the bass line involved by creating inversions.</li>
<li>If your melody line moves from the 3rd of a chord to an upper root of the chord, you can mirror that motion by moving the bass from the root of the chord up to the 3rd. It creates interesting interplay between melody and bass.</li>
</ol>
<p>There are times when you need to be careful about using inversions:</p>
<ol>
<li>If the melody and bass line are on the same note, don&#8217;t invert the next chord if it means both melody and bass move to the same note (i.e., avoid parallel octaves).</li>
<li>Be careful about leaping to inversions if there seems to be no good need or reason. For example, this might be hard to make work: C  F/A  Dm  G/B&#8230; etc. These inversions will often result in the ear being confused.</li>
<li>Avoid inversions if it means that the bass note following the inversion is an augmented 4th or diminished 5th away. In other words, this will be a bit hard on the ear: C  G/B  F, because the bass moving from the note B to F will be somewhat unpleasant.</li>
</ol>
<p>Inversions can be a great way to make an otherwise boring progression a bit more interesting, and can get some motion in your bass line, but as with most elements at your disposal, be careful how you use it.<br />
________________</p>
<p><a href="http://www.secretsofsongwriting.com/prepurchase2.html"><img class="alignleft size-thumbnail wp-image-993" title="Songwriting E-books by Gary Ewer" src="http://garyewer.wordpress.com/files/2009/11/all_6_refl10.jpg?w=120" alt="" width="120" height="150" /></a>Gary Ewer&#8217;s Songwriting E-books can clear up the mystery of chord progressions, melodies, lyrics and other vital elements of your songs. If you&#8217;re ready to unleash your songwriting potential, <strong><a href="http://www.secretsofsongwriting.com/prepurchase2.html">click here to read more.</a></strong></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Moció del Grup ICV-EUiA per la inclusió de clausules socials als projectes del nou Fons Estatal d'Inversió Local ]]></title>
<link>http://stcvo.wordpress.com/2009/11/18/mocio-del-grup-icv-euia-per-la-inclusio-de-clausules-socials-als-projectes-del-nou-fons-estatal-dinversio-local/</link>
<pubDate>Wed, 18 Nov 2009 18:39:35 +0000</pubDate>
<dc:creator>Jaume Massanés i Papell</dc:creator>
<guid>http://stcvo.wordpress.com/2009/11/18/mocio-del-grup-icv-euia-per-la-inclusio-de-clausules-socials-als-projectes-del-nou-fons-estatal-dinversio-local/</guid>
<description><![CDATA[Vist que l’equip de govern de l’ajuntament de Sant Cugat volia dedicar tots els diners del  nou Fons]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p style="text-align:justify;"><a href="http://stcvo.wordpress.com/files/2009/11/nueva-imagen.png"><img class="alignleft size-medium wp-image-362" title="Nueva imagen" src="http://stcvo.wordpress.com/files/2009/11/nueva-imagen.png?w=300" alt="" width="268" height="167" /></a>Vist que l’equip de govern de l’ajuntament de Sant Cugat volia dedicar tots els diners del  nou Fons Estatal d’Inversió Local (8,5 milions d’euros) a inversions.</p>
<p style="text-align:justify;">Vista la poca transparència i dubtosa prioritat o necessitat d’alguns dels projectes de l’anterior F.E.I.L. davant d’altres obres mes justificades i urgents.</p>
<p style="text-align:justify;">Iniciativa va negociar amb l’equip de govern perquè reconsiderés la seva postura inicial aconseguint consensuar i aprovar  finalment la següent moció:</p>
<p style="text-align:justify;"><em><span style="color:#800000;">Atesa la situació de crisi econòmica que es produeix a tot el país i també a Sant Cugat del Vallès.</span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">Atès que el govern estatal torna a posar en marxa el Fons Estatal d’inversió Local per dinamitzar l’economia dels municipis. </span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">Atès que la principal finalitat del fons és l’ocupació, tant de nova creació com el manteniment de llocs de treball. Els nous contractes hauran de realitzar-se a treballadors inscrits com a demandants d’ocupació amb prioritat per als aturats de llarga durada (més de dotze mesos registrats).</span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">Atès que el fons beneficiarà al teixit empresarial, especialment a les empreses, sobretot les petites i mitjanes, centrades en les activitats vinculades a la sostenibilitat econòmica, mediambiental i social que es troben a Sant Cugat del Vallès.</span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">Atès que el fons beneficiarà a l&#8217;ajuntament de Sant Cugat del Vallès que podrà finançar inversions i actuacions en benefici de la ciutadania per valor de 8,5 milions d’€.</span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">L’Ajuntament de Sant Cugat pren els següents acords: </span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">1.- Es prioritzarà la contractació de treballadors i treballadores en situació d’atur de llarga durada (que portin més de 12 mesos).</span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">2.- Els treballadors i treballadores contractats seran, prioritariament, treballadors inscrits al Servei Municipal d’Ocupació com a demandants d&#8217;ocupació a Sant Cugat del Vallès.</span></em></p>
<p style="text-align:justify;"><em><span style="color:#800000;">3.- Es donarà compte al Ple Municipal de la relació final de projectes que es desenvolupin pel Pla FEIL a Sant Cugat del Vallès</span></em>.</p>
<p style="text-align:right;"><em><span style="color:#800000;">Sant Cugat del Vallès, novembre de 2009</span></em></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Loads of Blackest Night!]]></title>
<link>http://mechanisticmoth.wordpress.com/2009/11/14/loads-of-blackest-night/</link>
<pubDate>Sun, 15 Nov 2009 05:48:13 +0000</pubDate>
<dc:creator>MechanisticMoth</dc:creator>
<guid>http://mechanisticmoth.wordpress.com/2009/11/14/loads-of-blackest-night/</guid>
<description><![CDATA[Did you know that there&#8217;s a lot of Blackest Night comics that I haven&#8217;t reviewed?  I mea]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p style="text-align:center;"><img class="aligncenter" src="http://fourcolors.blogs.pressdemocrat.com/files/2009/08/blackest_night_by_sinccolor1.jpg" alt="" width="512" height="348" /></p>
<p>Did you know that there&#8217;s a lot of Blackest Night comics that I haven&#8217;t reviewed?  I mean&#8230; really.  I guess it&#8217;s time to play catch up on our fun mega event of DC.</p>
<p>There&#8217;s a lot of zombie bashing both positive and negative after the click.</p>
<p><!--more--></p>
<p style="text-align:center;"><img src="http://dchallofjustice.com/news/wp-content/uploads/2009/10/batman-blackest-night-3.jpg" alt="" width="170" height="283" /><a id="zoomedLink" title="Click to zoom out." href="void(0);"><img src="http://i652.photobucket.com/albums/uu244/Godmera/blackest-night-event-20090717051719.jpg" alt="blackest-night-event-20090717051719.jpg image by Godmera" width="170" height="282" /></a><img src="http://www.dccomics.com/media/product/1/3/13175_400x600.jpg" alt="" width="169" height="281" /></p>
<p>So, the first batch of miniseries tying around the Blackest Night event ended within the last couple of weeks.</p>
<p>I have to say that really&#8230; it wasn&#8217;t too much of a success.  Sure, some parts were nice like seeing Deadman for a bit&#8230; and then not&#8230; and then having Dove suddenly becoming a surprising (even to herself) player in the event.  But, overall, it just didn&#8217;t all flow to me.  I think part of it was just that it was all too ridiculous for me. </p>
<p>Batman was a big stupid stupid fest of stupidness in the last issue where The Flying Graysons and Tim Drake&#8217;s dad recreate their deaths in an incredibly elaborate way to get some sort of emotions out of the Big B and Red R.  Then, the whole Mr. Freeze gun/last ditch effort to hide themselves from the Black Lanterns was way too contrived.</p>
<p>Superman was just a big brawl fest.  Plus, the characters were pretty flat.  Sure, it was nice to learn that Superboy&#8217;s arm could have fear in it, and to see the Psycho Pirate back (and then his semi-useful mask used as a tool against the Black Lanterns&#8230; which crumbles in two frames which took a lot of rereading to figure out).  Overall, the miniseries just accomplished me feeling really bad for Smallville.  Poor one-dimensional stereotypical hickish townsfolk&#8230;</p>
<p>Titans was a love affair, hate affair, okay affair.  Part of me could see past the T&#38;A fest when the story was okay, but then the other part of me was super offended.  I like Dove&#8217;s new role, but that&#8217;s about it.  Plus, it would have been cool to see Raven in this&#8230; but she wasn&#8217;t there.  Oh well.</p>
<p>Overall, the first batch of miniseries get an okay out of me&#8230; not really satisfying but not really terrible&#8230; they just don&#8217;t equal the power in the main feature.</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/25StarRatingscopy.jpg?t=1258260981" alt="25StarRatingscopy.jpg picture by PseudoPsychic" /></p>
<p>&#160;</p>
<p>The next grouping is the first two weeks of the ring promotional deal!  You know, when this was first announced (you&#8217;d get a promotional plastic ring with one of the Lantern Corps symbol on it for buying a certain comic) I thought it was a great marketing ploy but I wasn&#8217;t too excited to buy some comics that I wouldn&#8217;t normally get.  Well, each one of them proved me wrong in their own little ways.</p>
<p><strong>Doom Patrol #4:</strong></p>
<p style="text-align:center;"><img class="aligncenter" src="http://i.newsarama.com/dcnew/Nov09/1/dpatrol_4.jpg" alt="" width="259" height="368" /></p>
<p>Now, I&#8217;ve been meaning to read some of the classic Doom Patrol stories, but I just haven&#8217;t had the time.  I know there&#8217;s a giant mythos behind the group, but I can honestly say that I don&#8217;t really know any of it.  Now, all of the promotional books (thus far) have included a memory download for the Black Lantern ring to sort of bring new readers (as in the people just buying the comic for the ring) up to speed.  I don&#8217;t really feel that this one completely accomplished that unlike the other two.  Sure, it definitely helped a hell of a lot, but not completely.  Part of me liked the art in this comic and part of me didn&#8217;t&#8230; It looked good in some instances (most notably how the Black Lanterns have sort of a mummified look to them and the representation of emotions through the Black Lantern&#8217;s eyes now used as a common device in all of the Blackest Night books), but bads in others (I didn&#8217;t too much like the inking or coloring, really).  The reveal at the end was super interesting with Clifford Steele and has me considering picking up the next issue.  To tell you the truth, I didn&#8217;t read the back up feature&#8230; I really like J.M. Dematteis especially after Savior 28, but to have a back up feature and then a preview for an upcoming book sort of steals away from the flow.</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/3StarRatingscopy.jpg?t=1258261539" alt="3StarRatingscopy.jpg picture by PseudoPsychic" /></p>
<p>&#160;</p>
<p><strong>R.E.B.E.L.S. #10</strong></p>
<p style="text-align:center;"><img class="aligncenter" src="http://i.newsarama.com/dcnew/Nov09/1/rebels10.jpg" alt="" width="234" height="343" /></p>
<p>First things first, I really loved the art in this comic.  The highly stylized use of a thin line and the super cool inking with dashed lines to show texture (especially on faces) was just way too neat.  The facial expressions are over the top and yet spot on.</p>
<p>Anyway, this issue follows Braniac 2 and 3&#8230; I knew a little bit about the different Braniacs, but the Memory Download beginning was a nice refresher.  My question was why does Vril Drox (Braniac 2) still hang out with a giant posse (including Kanjar Ro) when he has robots to do his shit for him?</p>
<p>Oh well, the reveal at the end was priceless and shows that something done before in the Sinestro Corps War can be done again and still have a kickass effect.</p>
<p>Plus, Black Lantern Harbinger&#8217;s in here&#8230; and she&#8217;s not that whiny!</p>
<p>I&#8217;m definitely considering picking up the next one, and this review&#8217;s sort of sparse because the comic mostly speaks for itself.</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/35StarRatingscopy.jpg?t=1258261857" alt="35StarRatingscopy.jpg picture by PseudoPsychic" /></p>
<p>&#160;</p>
<p><strong>Booster Gold #26</strong></p>
<p style="text-align:center;"><img class="aligncenter" src="http://www.dccomics.com/media/product/1/3/13357_400x600.jpg" alt="" width="226" height="328" /></p>
<p>Now, I have a love/hate relationship with Booster Gold&#8230; He&#8217;s a showboat who&#8217;s now on the right track, but that still doesn&#8217;t forgive for his past mistakes of being sort of a stupid character.  Like everyone else (mostly), I feel like 52 definitely vindicated him a lot (which makes me think&#8230; are we ever going to see how Animal Man is dealing with Blackest Night?), but there&#8217;s still some problems in his comic now.</p>
<p>Despite having an unbalanced relationship with Booster Gold, I&#8217;ve always liked Ted Kord Blue Beetle&#8230; mostly because the first issue I read with him in it was Infinite Crisis and him being all badass with Max Lord and getting his brains blown out.</p>
<p>Overall, I came out of reading this comic saying aloud &#8220;Holy shit! That was a lot better than I thought it would be.&#8221;</p>
<p>From Booster revisiting one of his failures to the relationship with the new Blue Beetle, I really thought this issue excelled, and there&#8217;s no doubt that I will buy the next comic.  Part of the reason this comic works so well is that we know Ted Kord&#8217;s going to come bursting in at some point, but Dan Jurgens holds it off until the end and instead works on developing/reminding the reader of Booster and BBs friendship and characters.  That makes the sickening reveal at the end so much better.</p>
<p>Does anyone else find that Ted Kord&#8217;s nose looks really weird and a little uncharacteristic of him in life despite being all decaying&#8230;?  Also, couldn&#8217;t they have made his costume on the head sink in just a bit to represent the head wound&#8230;?</p>
<p>Nevertheless,</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/4StarRatingscopy.jpg?t=1258262401" alt="4StarRatingscopy.jpg picture by PseudoPsychic" /></p>
<p>&#160;</p>
<p>Our next grouping is the actual main event players a la Green Lantern, Corps, and Blackest Night.</p>
<p><strong>Green Lantern #47</strong></p>
<p style="text-align:center;"><img class="aligncenter" src="http://xmages.net/out.php/i586256_GreenLantern47.jpg" alt="" /></p>
<p>You know, I really don&#8217;t care much about Hal Jordan anymore.  Sure, he&#8217;s cool, but he&#8217;s just a Batman in space with a ring&#8230;  The stars of this book are the other players: Sinestro, Larfleeze, and Atrocitus.  Holy shit do I still want them to stay around when this is all over.  They&#8217;ve had such little screen time (excluding Sinestro) that I really want them to play out for much longer.  Hell!  Atrocitus gets his heart ripped out by the Inversions (who look super neat) like everyone else and ends up still living because he doesn&#8217;t need a heart because he&#8217;s such a badass.</p>
<p>Too bad Abin Sur doesn&#8217;t stay around for a bit longer&#8230; I guess he&#8217;s more important for the ideal and the standard of being the best Green Lantern ever than actually an interesting character.</p>
<p>Unfortunately, this comic doesn&#8217;t accomplish what it&#8217;s meant to, and that&#8217;s making Green Lantern seem interesting&#8230; it&#8217;s a bit of a sign of bad tidings when your lead character&#8217;s outshined by others.</p>
<p>Still, a respectable:</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/35StarRatingscopy.jpg?t=1258262905" alt="35StarRatingscopy.jpg picture by PseudoPsychic" /></p>
<p>&#160;</p>
<p><strong>Green Lantern Corps #42</strong></p>
<p style="text-align:center;"><img class="aligncenter" src="http://www.dccomics.com/media/product/1/3/13355_400x600.jpg" alt="" width="259" height="369" /></p>
<p>Stuff on Oa increases the shit up&#8230; there&#8217;s a lot of characters and I feel like only Kilowog gets a decent amount of treatment.</p>
<p>Sure, Kyle Rayner&#8217;s sacrifice at the end is pretty neat, but after just seeing something similar with Sodom Yat, the effect is lessened.  Plus, in the whole Origins and Omens thing that came out to get people excited about Blackest Night, Kyle was all painting a mural&#8230; So, I know that this isn&#8217;t going to last.</p>
<p>I don&#8217;t know&#8230; this comic just sort of felt like a placeholder in the event to find out what&#8217;s happening elsewhere.  Hmm&#8230;</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/25StarRatingscopy.jpg?t=1258263174" alt="25StarRatingscopy.jpg picture by PseudoPsychic" /></p>
<p>&#160;</p>
<p><strong>Blackest Night #4:</strong></p>
<p style="text-align:center;"><img class="aligncenter" src="http://artisticallychill.com/blog/wp-images/blackestNight04.jpg" alt="" width="282" height="391" /></p>
<p>Do the Justice League villains on the front of this cover really play much of a role on the inside&#8230; no&#8230; but they look cool.</p>
<p>Wow, but isn&#8217;t the part in Gotham so neat&#8230; Scarecrow just walking around and stares down Black Lantern Azrael (who looks super cool with that hood even if it&#8217;s a bit like Black Lantern Hawk).</p>
<p>Then, Luthor&#8217;s being a badass.</p>
<p>Then, Mera is suddenly a power player with the Flash and Atom&#8230; the new Big Three.  Then, Jason fights Ronnie and makes me want to see what happens.  Then, Atom gives a pep talk that sort of back fires (sadly) but rips a guy apart.</p>
<p>Then, in two pages in creepy near silence, Nekron rises&#8230; as does all of Coast City.</p>
<p style="text-align:center;"><img class="aligncenter" src="http://i50.photobucket.com/albums/f328/PseudoPsychic/Star%20Ratings/45StarRatingscopy.jpg?t=1258263660" alt="45StarRatingscopy.jpg picture by PseudoPsychic" /></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Core Essentials in Yoga &amp; Life]]></title>
<link>http://yoganamastacyblog.wordpress.com/2009/11/03/core-essentials-in-yoga-life/</link>
<pubDate>Tue, 03 Nov 2009 20:55:58 +0000</pubDate>
<dc:creator>yoganamastacyblog</dc:creator>
<guid>http://yoganamastacyblog.wordpress.com/2009/11/03/core-essentials-in-yoga-life/</guid>
<description><![CDATA[Feel like you need a breakthrough in moving your yoga practice or life to the next level? Awakening ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Feel like you need a breakthrough in moving your yoga practice or life to the next level?<br />
Awakening your core is the key to untapped strength, unlocking your hips &#38; inversions.<br />
We can so easily be knocked off balance by reaching outside ourselves to fulfill the needs of others. In our yoga practice, strengthening our core connects us with balance, stamina, strength, energy &#38; power within &#8211; yet we get off our mat and begin grasping for balance by managing our time or juggling priorities. Reconnect with your core &#38; flourish in your yoga practice &#38; in life.<br />
The Core Essentials Workshop is designed to:<br />
* Ignite the Power in Your Core<br />
* Move Beyond Limitations<br />
* Find True Balance from within<br />
* Align with what works for you<br />
* Command Your Energy &#38; presence<br />
* Cultivate the benefits of Yoga with Core Life Practices<br />
This workshop is lead by myself &#38; Cindy Silbert, Author &#38; Transformational Guide.<br />
Date: Saturday, November 21st<br />
Time: 2:30 -4:30pm<br />
Cost: $39 Sign up before November 16th<br />
          $49 After November 16th<br />
Email us at Stacy@yoganamastacy.com or Cindy@BringUtoLife.com for more information.<br />
<a href="http://www.yoganamastacy.com/coreessentials/"><a href="http://www.yoganamastacy.com/coreessentials/"></a><a href="http://www.yoganamastacy.com/coreessentials/"></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Legs up the Wall Pose]]></title>
<link>http://stilllakeyoga.wordpress.com/2009/10/29/legs-up-the-wall-pose/</link>
<pubDate>Thu, 29 Oct 2009 18:57:23 +0000</pubDate>
<dc:creator>stilllakeyoga</dc:creator>
<guid>http://stilllakeyoga.wordpress.com/2009/10/29/legs-up-the-wall-pose/</guid>
<description><![CDATA[It’s the end of the day, and your legs feel heavy, tired and achy. Maybe your work demands that you’]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>It’s the end of the day, and your legs feel heavy, tired and achy.</p>
<p>Maybe your work demands that you’re on your feet all the day or that you’re confined for hours to a car or truck.  Or maybe you’re vacationing and are stuck for an abominable length of time in a cramped airline seat.  Perhaps you’ve spent a fun-filled day toddling around Disney World but have stood in oh-so-many lines. </p>
<p>Whether at work or play, you might finish the day with legs just begging for a break.  Try the yoga Rx for tired legs:  Legs up the Wall Pose.</p>
<p>Getting into this revitalizing pose can be tricky, but it gets easier with practice.  Grab a mat, blanket, or towel (a great hotel room choice) and head to an expanse of wall.  Lay your “mat” on the floor touching the wall and stretching out away from it.  Sit on the mat with your knees bent, thighs drawn in towards the chest.  Have the right side of your body contacting the wall. </p>
<p>As you drop down onto your right elbow, pivot and swing your legs onto the wall.  As the legs draw up, the torso and head release to the floor.  Your legs should be straight and your feet about a foot apart.  By using your arms, you may be able to shimmy in and get your hips to contact the wall.  However, if the muscles in the backs of your legs are tight, it’s fine to keep the hips further from the wall so the legs can straighten.  Make adjustments to ensure the pose is symmetrical. </p>
<p>If you like, prop your head up slightly on a small pillow.  Rest your hands on the floor with your palms facing up.  Now, simply relax and let gravity do the work.  Allow all fatigue and heaviness in the legs to vanish.  You’re experiencing a great boon to your circulation and helping to prevent varicose veins.  </p>
<p>Feel your entire back melt to the floor.  A release of lower back tension is another benefit of the posture.  Remain in Legs up the Wall Pose for five to fifteen minutes.  Enjoy some relaxing music or simply follow the ebb and flow of your breath. </p>
<p>To come out of the pose, draw your knees to your chest, reach your right arm over your head, and roll onto your right side.  Slowly press yourself up.</p>
<p> Day after day, your legs do so much for you.  Treat them to Legs up the Wall Pose often. </p>
<p>Note:  Those with serious eye problems or back problems or high blood pressure should check with their physician before practicing any inverted (upside down) yoga postures.</p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Ja s’han iniciat les obres del Celler. Per fi!]]></title>
<link>http://rogergarcia.wordpress.com/2009/10/04/ja-s%e2%80%99han-iniciat-les-obres-del-celler-per-fi/</link>
<pubDate>Sun, 04 Oct 2009 15:42:49 +0000</pubDate>
<dc:creator>rogergarcia</dc:creator>
<guid>http://rogergarcia.wordpress.com/2009/10/04/ja-s%e2%80%99han-iniciat-les-obres-del-celler-per-fi/</guid>
<description><![CDATA[Recordeu quan va es va enfonsar la teulada del Celler?, l’agost del 2007. M’acabava d’estrenar com a]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><div class="separator" style="clear:both;text-align:center;"><a style="clear:left;float:left;margin-bottom:1em;margin-right:1em;" href="http://1.bp.blogspot.com/_KEaE517SiGs/SsiJeURFpKI/AAAAAAAADFI/4b4ex79-1KE/s1600-h/celler_columnes_g.gif"><img src="http://1.bp.blogspot.com/_KEaE517SiGs/SsiJeURFpKI/AAAAAAAADFI/4b4ex79-1KE/s200/celler_columnes_g.gif" border="0" alt="" /></a></div>
<div style="text-align:justify;">Recordeu quan va es va enfonsar la teulada del Celler?, l’agost del 2007.<br />
M’acabava d’estrenar com a regidor. L’edifici del <a href="http://ca.wikipedia.org/wiki/C%C3%A8sar_Martinell_i_Brunet">Cèsar Martinell</a> (1888-1973), l’arquitecte de les Catedrals del Vi!  Va ser una molt mala notícia d’estiu i no sabíem quin seria l’abast de la tragèdia. Pensàvem que podia caure una part del mur de la façana. Per sort tot va quedar en un ensurt ja que la teulada enfonsada no formava part de l’obra original.</div>
<div style="text-align:justify;">A partir d’aquí vam fer un estudi en profunditat de l’edifici i vam concloure que calia reforçar una fonamentació molt escassa del Celler per assegurar la seva estabilitat. Era el moment de rehabilitar l’edifici per permetre que pugui ser visitat.</div>
<div style="text-align:justify;">I desprès d’aconseguir una subvenció del Ministeri de Foment d’uns 700.000€ a més del pressupost municipal, vam encarregar el projecte executiu a l’arquitecte especialista en edificis històrics, Joan Albert Adell. Ja s’han adjudicat les obres i podem iniciar la primera fase que afecta a les estructures (fonamentació i teulada) i a la rehabilitació interior, fent que es pugui disposar d’una gran sala al soterrani i que tot l’edifici es pugui visitar i gaudir dels seus elements arquitectònics.</div>
<div style="text-align:justify;">La segona fase, lligada a la primera, servirà per fer els banys i rehabilitar els porxos exteriors, i per assegurar que compleixi la normativa requerida a un edifici públic (accessibilitat, instal•lacions&#8230;.).</div>
<div style="text-align:justify;">Ara. a més, hem obtingut una subvenció dels fons europeus de 500.000€ que ens permetrà poder licitar aquesta segona fase sense problemes.</div>
<div style="text-align:justify;">Rubí es mereix poder gaudir de la nostra <a href="http://ca.wikipedia.org/wiki/Catedral_del_vi">Catedral del Vi</a>. Un edifici de gran valor arquitectònic i la memòria històrica del passat vinícola de la ciutat. Com diu la dita, “<em>Rubí noies maques i bon vi</em>”.<br />
Amb la rehabilitació del Celler podrem recordar el procés de producció del vi rubinenc.</div>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[O BNG anuncia que a Deputación embargará 680.000 euros a Ponteareas do plan de obras 2009-2011]]></title>
<link>http://apeneiradocondado.wordpress.com/2009/09/23/o-bng-anuncia-que-a-deputacion-embargara-680-000-euros-a-ponteareas-do-plan-de-obras-2009-2011/</link>
<pubDate>Wed, 23 Sep 2009 10:02:16 +0000</pubDate>
<dc:creator>apeneiradocondado</dc:creator>
<guid>http://apeneiradocondado.wordpress.com/2009/09/23/o-bng-anuncia-que-a-deputacion-embargara-680-000-euros-a-ponteareas-do-plan-de-obras-2009-2011/</guid>
<description><![CDATA[Por resolución do Presidente Rafael Louzán, a Deputación Provincial acaba de detraerlle ao Concello ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Por resolución do Presidente Rafael Louzán, a Deputación Provincial acaba de detraerlle ao Concello de Ponteareas 680.802,74 euros, así o adianta o BNG de Ponteareas. <!--more-->Este importe, segundo explican os nacionalistas, é deducido do Plan de inversións da Deputación  para o perído 2009-2011 que sumaba 1.208.545,23 euros. Co embargo deste importe, Ponteareas non vai investir nada en obras e servizos básicos para os veciños con cargo a partida do presente ano e únicamente  251.503,58 no 2010 e 276.238,91 no 2011, só un 43,6% do total, puntualizan.</p>
<p>O BNG recorda que recentemente o Presidente da Deputación visitou o Concello e anunciou conxuntamente co Alcalde que en Ponteareas o investimento deste plan anti-crisis da Deputación superaba o millón douscentos mil euros, cando xa estaba asinada a retención dos 680.802,74 euros. &#8220;Louzán e Solla reduciron o plan únicamente a propaganda. Todo un exemplo de cinismo político e de falta de respeto a Institución Municipal e aos veciños e veciñas de Ponteareas”, manifestou Xosé Represas, Portavoz do BNG.</p>
<p>A débeda que agora a Deputación lle detrae ao Concello foi xerada dende o ano 2005, coincidindo co período que Salvador G. Solla leva de Alcalde. &#8220;Agora, que Salvador G. Solla tamén forma parte do goberno do PP na Deputación, ésta embargalle ao Concello 680.000 euros&#8221;.</p>
<p>Os nacionalistas tamén acusan a Nava Castro de &#8220;mentir descaradamente ante o Pleno da Corporación&#8221;. Recordan que como concelleira de Facenda, a requerimento do BNG, afirmaba hai poucos meses que o Concello non tiña débedas coa Deputación a 31 de decembro de 2008. Na resolución do Presidente da Deputación de 24 de agosto de 2009 recollese, “se detrae a cantidade de 680.802,74 euros por débedas dos anos 2005,2006,2007,2008 e 2009”</p>
<p>Para Represas “a crítica situacion económica do concello, agravada aínda máis nos anos que Salvador G. Solla leva de alcalde, ten colocado a Ponteareas nunha situación límite na que por tamén se perden os investimentos da Deputación. Como consecuncia da xestión de Salvador G. Solla e agora co embargo da Deputación, os veciños quedan privados de obras de infraestructura básicas como é o saneamento&#8221;, manifestou o nacionalista.</p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Un edifici sostenible i construït en temps record, la nova escola bressol]]></title>
<link>http://rogergarcia.wordpress.com/2009/09/16/un-edifici-sostenible-i-construit-en-temps-record-la-nova-escola-bressol/</link>
<pubDate>Wed, 16 Sep 2009 14:37:39 +0000</pubDate>
<dc:creator>rogergarcia</dc:creator>
<guid>http://rogergarcia.wordpress.com/2009/09/16/un-edifici-sostenible-i-construit-en-temps-record-la-nova-escola-bressol/</guid>
<description><![CDATA[Fa uns dies l’Alcaldessa va inaugurar l’Escola Bressol Municipal Bruna, un edifici construït en mig ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img src="http://rogergarcia.wordpress.com/files/2009/09/bressol2g.gif?w=150" alt="bressol2g" title="bressol2g" width="150" height="86" class="alignleft size-thumbnail wp-image-122" /></p>
<p style="text-align:justify;">Fa uns dies l’Alcaldessa va inaugurar l’Escola Bressol Municipal Bruna, un edifici construït en mig any i que pot marcar una línia en quan a tipologia d’edificis escolars a la ciutat<br />
Es tracta d’un edifici original, funcional i integrat a l’entorn que està dissenyat amb criteris bioclimàtics amb una sistema de climatització natural i fent servir l’energia solar<br />
Si us fixeu veureu que està orientat al sud i que  compta amb il·luminació natural a totes les aules, tant des del pati com pel sostre. El sistema d’il•luminació elèctric es regula en funció de la intensitat lumínica exterior.<br />
A més, per aïllar les aules  s’ha construït una coberta verda amb cultiu extensiu i que com diuen els autors del projecte es convertirà en la cinquena façana. I en aquesta línia, s’ha construït uns  voladissos que permeten que els raigs del sol arribin directament a les aules a l’hivern, mentre que eviten la radiació directa els mesos d’estiu<br />
Compta amb uns mecanismes de ventilació però sense necessitat de fer servir suports calorífics o frigorífics extraordinaris i l’energia solar que s’ha instal·lat serà per a l’aigua sanitària i la calefacció. Els radiadors són de baix consum i gran rendiment calòric que al no sobrepassar els 42 graus eviten que els nens es puguin cremar.</p>
<p style="text-align:justify;">Però a més, aquest edifici s’ha construït en un temps record. Necessitàvem que l’edifici estigues acabat per l’inici del curs escolar i ho hem aconseguit, entre d’altres factors, escollint el sistema de  concurs de “projecte + obra”en el que mitjançant un concurs  d’idees previ es selecciona conjuntament un equip d’arquitectes, un projecte i una empresa constructora en un sol paquet,  d’aquesta manera vam guanyar alguns mesos del període de licitacions dels projectes de les obres.<br />
Ara ja el tenim en funcionament. Un edifici amb un disseny que l’integra a l’entorn i que com va dir l’Alcaldessa, contribueix a la millora del barri de la Plana de Can Bertran, juntament amb la nova biblioteca i amb la nova plaça i l’ampliació del  parquing del Mercat.</p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[blah blah yoga blah yoga blah blah]]></title>
<link>http://crisitunity.wordpress.com/2009/09/13/blah-blah-yoga-blah-yoga-blah-blah/</link>
<pubDate>Sun, 13 Sep 2009 12:50:12 +0000</pubDate>
<dc:creator>crisitunity</dc:creator>
<guid>http://crisitunity.wordpress.com/2009/09/13/blah-blah-yoga-blah-yoga-blah-blah/</guid>
<description><![CDATA[I am still high from teaching yesterday. When BF came home, he hugged me and said &#8220;Hey there, ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>I am still high from teaching yesterday. When BF came home, he hugged me and said &#8220;Hey there, yoga teacher.&#8221; He makes me smile inside and out.</p>
<p>I did a home practice yesterday afternoon that was interesting. My intention was to work up to <a href="http://yoga.about.com/od/yogaposes/a/kingpigeon.htm">king pigeon pose</a>, but when I was warmed up enough, my body just wasn&#8217;t interested in getting into the pose. It was weird, because prior to now I have always, always, always been happy to add backbends to any practice I was doing. But not yesterday. I just wanted to work with my legs.</p>
<p>So that&#8217;s what I did. My <a href="http://yoga.about.com/od/yogaposes/a/hanumanasana.htm">hanuman</a> is coming along nicely; I&#8217;m not even six inches away from a full split. I did <a href="http://yoga.about.com/od/yogaphotogalleries/ig/Inversions-Photo-Gallery/Forearm-Stand.htm">forearm stand</a> for a while, as well, and yet again I find that all I need to do to feel my body come to the pose is to practice. Forearm stand has always felt compressed and not a little dangerous, but I worked with Paul and Jennifer on it a few times in the last couple of months. The biggest difference was made when I stopped caring about whether my head was in the right place; when I just looked behind me instead of down at the floor or forward, the curve in my upper back seemed less out of control, and my shoulders seemed to have more space.</p>
<p>The problem with any inversion, to me, is that your brain is telling you HOLY SHIT I&#8217;M UPSIDE DOWN AND I TOTALLY CAN&#8217;T BALANCE ON MY HANDS/FOREARMS/HEAD the entire time you&#8217;re doing it. This is very distracting. It makes no real sense, because when you come to think about balancing on your <em>feet</em> all day, it&#8217;s really not any more miraculous to balance on your forearms. You&#8217;ve just been balancing on your feet your whole life, and are accustomed to the weight of your body being distributed and balanced in one direction, rather than the other.</p>
<p>If I could get past the immediate mental block of HOLY SHIT I&#8217;M UPSIDE DOWN etc., I could focus on the finer aspects of the inversion and probably do it for a lot longer. My body just seems so <em>heavy</em> from the other direction; 132 pounds resting on my forearms seems like an awful lot. Even though the same 132 pounds rests easily on the tiny square footage of my feet every single day! I don&#8217;t know how to remind myself of this fact while I&#8217;m upside down; if I could get it through my head at that time, I think I could try to focus on feeling lightness in my arms and legs, or focus on allowing the muscles to do the work rather than just letting dead weight rest in my forearms. But my brain is in robot mode, saying, HEY THERE, YOU&#8217;RE UPSIDE DOWN. HAVE I MENTIONED YOU&#8217;RE UPSIDE DOWN? DUDE, YOU&#8217;RE TOTALLY UPSIDE DOWN.</p>
<p>I also found something interesting in <a href="http://yoga.about.com/od/yogaphotogalleries/ig/Standing-Poses-Photo-Gallery/Ardha-Chandrasana.htm">half-moon</a>. It&#8217;s a pose that has always been <a href="http://crisitunity.wordpress.com/2009/07/24/asana-ardha-chandrasana-half-moon-pose/">difficult</a> for me, and even though it has evolved as my practice has, I still find it exceedingly unstable. A sharp breeze could cause me to tumble to the ground when I&#8217;m in it. I&#8217;m not sure I really feel openness in my chest every time I do it, either; I&#8217;m sometimes twisted towards the ground, instead.</p>
<p>Yesterday I realized that the position of each non-weight-bearing limb has a major bearing on where the center of gravity is in the pose. I knew that if I moved my up leg further behind my body that it would change where my balance focused, but I found that moving my lower arm AND my upper arm forward and back makes a pretty big difference. I can&#8217;t think of a mechanical metaphor that fits this situation &#8211; propellers on a helicopter, maybe, or the weights in a clock &#8211; but I had to find a happy medium between keeping my arm forward and my leg back, and vice versa. Experimenting with this was quite a challenge because I kept almost falling out every time I made a tiny adjustment. But it was interesting.</p>
<p>I&#8217;ve also been doing this thing where I experiment with king dancer pose while I&#8217;m supporting myself, rather than treating it as the standing balance that it is. There are just <a href="http://images.google.com/images?hl=en&#38;source=hp&#38;q=king%20dancer%20pose&#38;um=1&#38;ie=UTF-8&#38;sa=N&#38;tab=wi">so many hugely different expressions</a> of that pose, and I want to fiddle with where I have my back and my head and my leg without worrying about whether I&#8217;m going to fall over. It&#8217;s something I actually want to bring to a class one day &#8211; the centerpiece being experimentation by the wall, rather than just the accomplishment of the pose on its own.</p>
<p>Full yoga post today. Sorry about that. I am feeling my yoga addiction quite strongly right now. Part of the reason is that I go to Santa Barbara in LESS THAN A MONTH. Eeeek!</p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Up the Wall and Off the Wall]]></title>
<link>http://greenspell.wordpress.com/2009/09/08/up-the-wall-and-off-the-wall/</link>
<pubDate>Tue, 08 Sep 2009 17:56:43 +0000</pubDate>
<dc:creator>Yoga Witch</dc:creator>
<guid>http://greenspell.wordpress.com/2009/09/08/up-the-wall-and-off-the-wall/</guid>
<description><![CDATA[If you love yoga as much as I do, I hope you have checked out EcoYogini&#8217;s Personal Practice Ad]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>If you love yoga as much as I do, I hope you have checked out <a href="http://ecoyogini.blogspot.com/2009/09/personal-practice-adventure-carving-out.html">EcoYogini&#8217;s Personal Practice Adventure</a>. I know, I know, it&#8217;s all I talk about lately &#8211; I&#8217;m just so excited about it! It came just as I was re-inspired in my yoga practice, so now I&#8217;m doubly inspired!</p>
<div id="attachment_1410" class="wp-caption alignright" style="width: 258px"><img class="size-full wp-image-1410" title="legsupwallpose" src="http://greenspell.wordpress.com/files/2009/09/legsupwallpose.jpg" alt="Photo from Yoga Journal" width="248" height="248" /><p class="wp-caption-text">Photo from Yoga Journal</p></div>
<p>For this adventure, <a href="http://greenspell.wordpress.com/2009/09/03/coming-together-in-yoga/">I have set a few intentions</a>. However, there is one more I wanted to add: I will do <a href="http://greenspell.wordpress.com/2009/05/16/permission-to-do-legs-up-the-wall/">Legs Up the Wall</a> at least four times a week for ten minutes at a time.</p>
<p>I have always had circulation problems. I have a long family history of heart malfunctions, varicose veins, and the like. There have been many days, even when I was a teenager, when, after standing for long periods of time, the pain in my legs was so intense that I would instinctively know that I had to reverse the blood flow. I often would lie on the floor and prop my legs up on the couch, which was really the only thing that helped the pain.</p>
<p>Little did I know that this was the beginning of Legs Up the Wall! Fast-forward to 2009, when I fail to do this pose regularly just because I think I don&#8217;t have 10 minutes to spare. Hmmmm&#8230; My Sivananda teacher often talks about this pose, challenging us to do it for ten minutes &#8211; every day. She says we often avoid it not just because of the time issue, but because it is, basically, an effortless pose, and we almost feel guilty for &#8220;indulging.&#8221; She said this is backwards thinking, as we should always strive to help our bodies be the best they can be, and any inversion does a great service to the body, no matter how &#8220;effortless&#8221; it may appear.</p>
<p>After spending the summer hiking (sometimes for <a href="http://greenspell.wordpress.com/2009/08/24/twelve-miles-and-seven-hours/">seven hours at a time</a>), I found that old, familiar leg pain has come back. I admit, I often ignored it, and did a few inversions when I could squeeze them in. But then I had some guidance that pushed me to make the commitment to do Legs Up the Wall on a regular basis.</p>
<p>A few weeks ago, my sister had a dinner party. One of her guests was a local intuitive healer, <a href="http://www.mariahcrawford.com">Mariah Crawford</a>. She is a wonderfully gifted woman, and I have gone to her for readings twice. When I told her I had to leave early due to painful cramps I was experiencing, she asked if I wanted her to do some Reiki on me. I have never had it done, even though both my sister and <a href="http://guidedpathhealing.webs.com/">one of my brothers</a> are Level II Reiki Healers, so I agreed. I was shocked by how much better it made me feel. During the impromptu session on my sister&#8217;s couch, Mariah said, &#8220;I&#8217;m receiving information that you are having stomach troubles and that the circulation in your legs is bad.&#8221; I happen to have been also experiencing bad leg pain that week, and terrible stomach aches (stress), and when she said &#8220;circulation&#8221; I knew it was time to commit to regular inversions!</p>
<p>So I am adding that to my intentions for the Personal Practice Adventure. Even as I write this, I am planning a moment before lunch to get my legs up the wall! They aren&#8217;t feeling so good right now, so I&#8217;m counting down the minutes until I can do this.</p>
<p>Lastly, speaking of walls, my computer is bouncing off them! It keeps signing me out of all my accounts and making it very difficult to get back into them. And equally annoying, it will not allow me to comment on other people&#8217;s blogs! I&#8217;ve been trying to leave comments at my favorite blogs all week, but to no avail. So please don&#8217;t think I&#8217;ve stopped reading your blogs &#8211; I&#8217;m still there! I&#8217;m hoping my boyfriend can fix this for me and then everything will be back to normal.</p>
<p>Until then &#8211; I&#8217;m putting my legs up the wall! <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_smile.gif' alt=':)' class='wp-smiley' /> </p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Counting Cycles of a Permutation in Parallel]]></title>
<link>http://rjlipton.wordpress.com/2009/09/04/counting-cycles-of-a-permutation-in-parallel/</link>
<pubDate>Fri, 04 Sep 2009 12:28:27 +0000</pubDate>
<dc:creator>rjlipton</dc:creator>
<guid>http://rjlipton.wordpress.com/2009/09/04/counting-cycles-of-a-permutation-in-parallel/</guid>
<description><![CDATA[How to count the number of cycles modulo in parallel Gian-Carlo Rota was one the world experts on co]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><font color="”#0066cc?"><br />
<em> How to count the number of cycles modulo <img src='http://l.wordpress.com/latex.php?latex=2+%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='2 &amp;fg=000000' title='2 &amp;fg=000000' class='latex' /> in parallel </em><br />
<font color="”#000000?"></p>
<p><img src="http://rjlipton.wordpress.com/files/2009/09/images.jpeg" alt="images" title="images" width="110" height="112" class="alignright size-full wp-image-3390" /></p>
<p>
Gian-Carlo Rota was one the world experts on combinatorics, and helped move the field from a corner of mathematics to become one of its central areas. Rota is famous for many other things, but his book <a href="http://www.amazon.com/Indiscrete-Thoughts-Gian-Carlo-Rota/dp/0817638660">Indiscrete Thoughts</a> is a classic&#8212;it is a great read. Even more famous, is his list of <a href="http://www.math.binghamton.edu/paul/comm-rota.pdf">Ten Lessons I Wish I Had Been Taught.</a></p>
<p>
Today I plan to follow one of Rota&#8217;s lessons and highlight a result that I have already presented. I have not done this before, but I think the result is sufficiently interesting to warrant this. Somehow it got lost before, since the post on it contained several other ideas and results. This time it will be alone. I hope you agree that it is worth repeating; I think it is an interesting &#8220;trick&#8221; that you might be able to use.<br />
<!--more--></p>
<p>
Rota&#8217;s ten lessons created a bit of &#8220;push-back&#8221; as people say. The lesson that got the most push-back was titled: </p>
<blockquote><p>
Publish the Same Result Several Times,
</p></blockquote>
<p>which caused many to get quite upset&#8212;perhaps more than a bit. This is, of course, the lesson that I am following today. </p>
<p>
Here is the exact quote from his paper:</p>
<p><p style="margin-right:5%;margin-left:5%;">
After getting my degree I worked for a few years in functional analysis. I bought a copy of Frederick Riesz&#8217;s <em>Collected Papers</em> as soon as the big, thick, heavy, oversize volume was published. However, as I began to leaf through, I could not help but notice that the pages were extra thick, almost like cardboard. Strangely, each of Riesz&#8217;s publications had been reset in exceptionally large type. I was fond of Riesz&#8217;s papers, which were invariably beautifully written and gave the reader a feeling of definitiveness.</p>
<p style="margin-right:5%;margin-left:5%;">
As I looked through his <em>Collected Papers</em>, however, another picture emerged. The editors had gone out of their way to publish every little scrap Riesz had ever published. It was clear that Riesz&#8217;s publications were few. What is more surprising is that the papers had been published several times. Riesz would publish the first rough version of an idea in some obscure Hungarian journal. A few years later he would send a series of notes to the French Academy&#8217;s <em>Comptes Rendus</em> in which the same material was further elaborated. A few more years would pass, and he would publish the definitive paper, either in French or in English.</p>
<p style="margin-right:5%;margin-left:5%;">
Adam Koranyi, who took courses with Frederick Riesz, told me that Riesz would lecture on the same subject year after year while meditating on the definitive version to be written. No wonder the final version was perfect.</p>
<p style="margin-right:5%;margin-left:5%;">
Riesz&#8217;s example is worth following. The mathematical community is split into small groups, each one with its own customs, notation, and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation and who will rightly claim it as his own. </p>
<p>
Let&#8217;s now turn to the result that I want to highlight, which was originally presented <a href="http://rjlipton.wordpress.com/2009/07/18/graphs-permutations-characters-and-logspace/">here.</a></p>
<p>
<p><b> Number of Cycles of a Permutation </b></p>
<p><p>
The result concerns permutations. Of course a permutation is nothing more than a mapping <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' /> from a finite set <img src='http://l.wordpress.com/latex.php?latex=%7BS%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{S}&amp;fg=000000' title='{S}&amp;fg=000000' class='latex' /> to itself that is one-to-one. The set is usually taken to be <img src='http://l.wordpress.com/latex.php?latex=%7B1%2C2%2C%5Cdots%2Cn%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1,2,\dots,n}&amp;fg=000000' title='{1,2,\dots,n}&amp;fg=000000' class='latex' /> for some <img src='http://l.wordpress.com/latex.php?latex=%7Bn%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{n}&amp;fg=000000' title='{n}&amp;fg=000000' class='latex' />, but we will allow any set <img src='http://l.wordpress.com/latex.php?latex=%7BS%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{S}&amp;fg=000000' title='{S}&amp;fg=000000' class='latex' />. </p>
<p>
The problem that I want to discuss is: does <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' /> have an even or an odd number of cycles? Every permutation can be uniquely written, up to order, as a collection of cycles. The number of cycles of a permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' /> will be denoted by <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7Bcycle%7D%28%5Cpi%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{cycle}(\pi)}&amp;fg=000000' title='{\mathsf{cycle}(\pi)}&amp;fg=000000' class='latex' />. </p>
<p>
Computing the number of cycles of a permutation seems to a <i>sequential</i> property. I am not exactly sure how to state this precisely, but the obvious algorithm requires that you pick a point <img src='http://l.wordpress.com/latex.php?latex=%7Bx+%5Cin+S%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x \in S}&amp;fg=000000' title='{x \in S}&amp;fg=000000' class='latex' />, and then trace out its cycle:
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle+x%2C+%5Cpi%28x%29%2C+%5Cpi%28%5Cpi%28x%29%29%2C+%5Cdots%2C+x%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle x, \pi(x), \pi(\pi(x)), \dots, x&amp;fg=000000' title='\displaystyle x, \pi(x), \pi(\pi(x)), \dots, x&amp;fg=000000' class='latex' /></p>
<p> Then, move on to the next cycle, and eventually discover the cyclic structure of <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' />. This yields, of course, the exact number of cycles of the permutation.</p>
<p>
There is another way to compute the number of cycles of a permutation modulo <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' /> that is more indirect. This is the trick that I want to highlight:</p>
<blockquote><p><b>Theorem: </b> <em> For any permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;bg=e8e8e8&amp;fg=000000' title='{\pi}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />,
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cmathsf%7Bcycle%7D%28%5Cpi%29+%5Cequiv+%26%23124%3BS%26%23124%3B+%2B+%5Cmathsf%7Binv%7D%28%5Cpi%29+%5Cbmod+2%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \mathsf{cycle}(\pi) \equiv &#124;S&#124; + \mathsf{inv}(\pi) \bmod 2&amp;bg=e8e8e8&amp;fg=000000' title='\displaystyle  \mathsf{cycle}(\pi) \equiv &#124;S&#124; + \mathsf{inv}(\pi) \bmod 2&amp;bg=e8e8e8&amp;fg=000000' class='latex' /></p>
<p> </em></p></blockquote>
<p> Here <img src='http://l.wordpress.com/latex.php?latex=%7B+%5Cmathsf%7Binv%7D%28%5Cpi%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{ \mathsf{inv}(\pi)}&amp;fg=000000' title='{ \mathsf{inv}(\pi)}&amp;fg=000000' class='latex' /> is the number of inversions of the permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' />: an inversion is a pair of <img src='http://l.wordpress.com/latex.php?latex=%7Bx%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x}&amp;fg=000000' title='{x}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7By%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{y}&amp;fg=000000' title='{y}&amp;fg=000000' class='latex' /> in <img src='http://l.wordpress.com/latex.php?latex=%7BS%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{S}&amp;fg=000000' title='{S}&amp;fg=000000' class='latex' /> so that <img src='http://l.wordpress.com/latex.php?latex=%7Bx%26%2360%3By%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x&lt;y}&amp;fg=000000' title='{x&lt;y}&amp;fg=000000' class='latex' /> and yet <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%28x%29+%26%2362%3B+%5Cpi%28y%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi(x) &gt; \pi(y)}&amp;fg=000000' title='{\pi(x) &gt; \pi(y)}&amp;fg=000000' class='latex' />. Here is an example: Consider the permutation that maps,
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++1+%5Crightarrow+2+%5Ctext%7B+and+%7D+2+%5Crightarrow+3+%5Ctext%7B+and+%7D+3+%5Crightarrow+1.%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  1 \rightarrow 2 \text{ and } 2 \rightarrow 3 \text{ and } 3 \rightarrow 1.&amp;fg=000000' title='\displaystyle  1 \rightarrow 2 \text{ and } 2 \rightarrow 3 \text{ and } 3 \rightarrow 1.&amp;fg=000000' class='latex' /></p>
<p> The number of cycles is <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' />, the cardinality of the set is <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' />, and there are <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' /> inversions. The inversions are <img src='http://l.wordpress.com/latex.php?latex=%7B1%2C3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1,3}&amp;fg=000000' title='{1,3}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7B2%2C3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2,3}&amp;fg=000000' title='{2,3}&amp;fg=000000' class='latex' />. The theorem works, since <img src='http://l.wordpress.com/latex.php?latex=%7B+1+%5Cequiv+3+%2B+2+%5Cbmod+2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{ 1 \equiv 3 + 2 \bmod 2}&amp;fg=000000' title='{ 1 \equiv 3 + 2 \bmod 2}&amp;fg=000000' class='latex' />.</p>
<p>
If we think of the cardinality of <img src='http://l.wordpress.com/latex.php?latex=%7BS%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{S}&amp;fg=000000' title='{S}&amp;fg=000000' class='latex' /> as known, then the theorem really says:
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Culcorner+%5Cmathsf%7Binv%7D%28%5Cpi%29+%5Cbmod+2+%5Curcorner+%5Ctext%7B+determines+the+value+of+%7D+%5Culcorner+%5Cmathsf%7Bcycle%7D%28%5Cpi%29+%5Cbmod+2+%5Curcorner.%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \ulcorner \mathsf{inv}(\pi) \bmod 2 \urcorner \text{ determines the value of } \ulcorner \mathsf{cycle}(\pi) \bmod 2 \urcorner.&amp;fg=000000' title='\displaystyle  \ulcorner \mathsf{inv}(\pi) \bmod 2 \urcorner \text{ determines the value of } \ulcorner \mathsf{cycle}(\pi) \bmod 2 \urcorner.&amp;fg=000000' class='latex' /></p>
<p> 	 The reason I am excited by the theorem is that computing <img src='http://l.wordpress.com/latex.php?latex=%7B+%5Cmathsf%7Binv%7D%28%5Cpi%29+%5Cbmod+2+%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{ \mathsf{inv}(\pi) \bmod 2 }&amp;fg=000000' title='{ \mathsf{inv}(\pi) \bmod 2 }&amp;fg=000000' class='latex' /> seems, to me, to be a much less sequential task, than the usual method of determining the cyclic structure of a permutation:</p>
<p>
<img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbullet%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\bullet}&amp;fg=000000' title='{\bullet}&amp;fg=000000' class='latex' /> The usual method is sequential. It follows a point to another point and eventually determines its cyclic structure. Of course this method determines more than just the number of cycles modulo <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' />: it gets the exact number of cycles. This method clearly is a sequential method, which does not seem to be parallelizable in any obvious manner.</p>
<p>
<img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbullet%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\bullet}&amp;fg=000000' title='{\bullet}&amp;fg=000000' class='latex' /> The inversion method is easily made parallel. Have each &#8220;processor&#8221; take a pair of <img src='http://l.wordpress.com/latex.php?latex=%7Bx%2Cy%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x,y}&amp;fg=000000' title='{x,y}&amp;fg=000000' class='latex' /> from <img src='http://l.wordpress.com/latex.php?latex=%7BS%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{S}&amp;fg=000000' title='{S}&amp;fg=000000' class='latex' />, and then check whether they form an inversion: that is <img src='http://l.wordpress.com/latex.php?latex=%7Bx%26%2360%3By%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x&lt;y}&amp;fg=000000' title='{x&lt;y}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%28x%29+%26%2362%3B+%5Cpi%28y%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi(x) &gt; \pi(y)}&amp;fg=000000' title='{\pi(x) &gt; \pi(y)}&amp;fg=000000' class='latex' />. This all can be done in one parallel step. Then, the processors each report back to a central processor, who computes the total number of inversions modulo <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' />.</p>
<p>
<p><b> Open Problems </b></p>
<p><p>
Can we generalize the inversion method to count cycles modulo something other than <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' />? Put another way: is there a parallel method for computing the number of cycles of a permutation modulo say <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' />? Note, simply computing the number of inversions modulo <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' /> will not allow the determination of the number of cycles modulo <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' /> of a permutation. That was one of the results of the earlier post. </p>
<p>
Finally, can we use the inversion method to solve an open problem? </p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[A Pedal Tone Adds Glue to a Wacky Chord Progression]]></title>
<link>http://garyewer.wordpress.com/2009/08/18/a-pedal-tone-adds-glue-to-a-wacky-chord-progression/</link>
<pubDate>Tue, 18 Aug 2009 17:20:10 +0000</pubDate>
<dc:creator>garyewer</dc:creator>
<guid>http://garyewer.wordpress.com/2009/08/18/a-pedal-tone-adds-glue-to-a-wacky-chord-progression/</guid>
<description><![CDATA[Written by Gary Ewer, from &#8220;The Essential Secrets of Songwriting&#8221; website. Follow Gary o]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Written by Gary Ewer, from &#8220;<strong><em><a href="http://www.secretsofsongwriting.com">The Essential Secrets of Songwriting</a></em></strong>&#8221; website.<br />
<a style="font-weight:bold;" href="http://www.twitter.com/garyewer" target="_blank"><span>Follow Gary on Twitter</span></a></p>
<table style="text-align:left;width:254px;height:1px;" border="0" cellspacing="1" cellpadding="1">
<tbody>
<tr>
<td style="background-color:#cc0000;"></td>
</tr>
</tbody>
</table>
<p>There are many ways to add a bit of variety to your chord choices. The problem is, the more you stray from the tried and true, the more confusing it can be to the listener. A <span style="font-weight:bold;font-style:italic;">pedal tone</span> is a great way to help the listener better understand a complicated progression.</p>
<p><img class="alignleft size-thumbnail wp-image-440" style="margin:5px;" title="multi-guit" src="http://garyewer.wordpress.com/files/2009/08/multi-guit.jpg?w=150" alt="multi-guit" width="150" height="99" />A pedal tone is a note, most often in the bass, that <span style="font-weight:bold;">stays constant while the chords above it change.</span></p>
<p>In a sense, a pedal tone acts like glue. As progressions become complex, that pedal tone gives the ear a landmark, of sorts, to latch on to. So even if the progression takes the listener far afield, <span style="font-weight:bold;">the pedal tone will give a sense of cohesion.</span> Notice, for example, progression #3 below: without the C pedal, that progression is a bit forced. But the C pedal makes better sense.</p>
<p>So if you&#8217;ve written a chord progression that you worry might be a bit too outside the norm, try adding a pedal tone. It works best if that bass pedal tone actually is the root of one of the chords in your progression (usually the first, or the last; but notice #5 below, where the G of the pedal never forms the root of a chord), but it can work with other chords, so try experimenting. And notice that for most of the chords in your progression, the pedal tone won&#8217;t even exist in the chord; that&#8217;s OK. Experiment until you find something that works.</p>
<p>Here are some examples (the note after the slash is the bass note):</p>
<p><span style="font-weight:bold;">1) C F/C G/C C</span><br style="font-weight:bold;" /><span style="font-weight:bold;">2) C/G F/G G C/G</span><br style="font-weight:bold;" /><span style="font-weight:bold;">3) C Ab/C Db/C G/C C</span><br style="font-weight:bold;" /><span style="font-weight:bold;">4) Am Em/A Am Dm/A Am</span><br style="font-weight:bold;" /><span style="font-weight:bold;">5) F/G Eb/G F/G Db/G Eb/G F/G</span></p>
<p><a href="http://www.secretsofsongwriting.com/prepurchase2.html"><br />
</a><span style="font-weight:bold;"><a href="http://www.secretsofsongwriting.com/prepurchase2.html"><img style="width:100px;height:92px;float:left;border:0 solid initial;" src="http://www.secretsofsongwriting.com/images/all_6_sm.jpg" alt="6 Songwriting E-books" /></a></span><span style="color:red;"><a href="http://www.secretsofsongwriting.com/prepurchase2.html">&#8220;The Essential Secrets of Songwriting&#8221;</a> shows you how to write great songs. It&#8217;s just one of a suite of 6 songwriting e-books written by Gary Ewer. (His</span><span style="color:red;"> newest e-book, &#8221;The Essential Secrets of Songwriting- Chord Progression Formulas&#8221; is being offered for free when you purchase any other of his songwriting e-books.) <span style="font-weight:bold;">Let these six e-books show you </span><span style="font-style:italic;font-weight:bold;">every aspect</span><span style="font-weight:bold;"> of</span></span><span style="color:red;font-weight:bold;"> <span style="font-style:italic;">how</span></span><span style="font-weight:bold;"><span style="color:red;"><span style="font-style:italic;"> to write great songs!</span> </span></span><a href="http://www.secretsofsongwriting.com/prepurchase2.html"><span style="font-weight:bold;">Read more..</span></a></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[How to Make a Chord Progression More Interesting]]></title>
<link>http://garyewer.wordpress.com/2009/08/06/how-to-make-a-chord-progression-more-interesting/</link>
<pubDate>Thu, 06 Aug 2009 14:37:02 +0000</pubDate>
<dc:creator>garyewer</dc:creator>
<guid>http://garyewer.wordpress.com/2009/08/06/how-to-make-a-chord-progression-more-interesting/</guid>
<description><![CDATA[Written by Gary Ewer Follow Gary on Twitter First, let me start by saying that most songwriters worr]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Written by Gary Ewer<br />
<a style="font-weight:bold;" href="http://www.twitter.com/garyewer" target="_blank"><span>Follow Gary on Twitter</span></a></p>
<table style="text-align:left;width:254px;height:1px;" border="0" cellspacing="1" cellpadding="1">
<tbody>
<tr>
<td style="background-color:#cc0000;"></td>
</tr>
</tbody>
</table>
<p><img class="alignleft size-full wp-image-419" style="margin:5px;" title="cpf_cover_s" src="http://garyewer.wordpress.com/files/2009/08/cpf_cover_s.jpg" alt="cpf_cover_s" width="130" height="179" />First, let me start by saying that most songwriters worry too much about their chord progressions. It&#8217;s actually not as important to get an interesting chord progression as it is to get one that actually works. But if you are looking for ways to get a bit of spice into an otherwisemundane progression, there are things you can easily do.</p>
<p>Try some of these ideas:</p>
<p><span style="font-weight:bold;">1) Use pedal bass.</span> This means placing a note in the bass (or the lowest sounding instrument), and keep that note there no matter what the chords do above it. It can change a boring progression into something much more interesting.<br />
For example: C F G C could become C F/C G/C C.</p>
<p><span style="font-weight:bold;">2) Use chord inversions.</span> Otherwise known as slash chords, an inverted chord places a note other than the root in the bass. This changes the sound of a chord in very subtle ways.<br />
For example: C G Am F C Dm C could become C G/B Am F C/E Dm C. These inversions allow the bass to move in a more melodic fashion, creating a second line of interest.</p>
<p><span style="font-weight:bold;">3) Use added tones.</span> An added tone simply means that you add a non-chord-tone to a basic triad.<br />
For example: C F G C could become Cadd9 Fadd9 G Cadd6. This will require experimentation to make it suit your song, but added tones are a great way to add an air of sophistication to your chord progression.</p>
<p>(If you&#8217;d like more chord progression ideas (modal mixtures, secondary dominants, etc.), try &#8220;<a href="http://www.secretsofsongwriting.com/prepurchase2.html">Essential Chord Progressions</a>&#8220;, &#8220;<a href="http://www.secretsofsongwriting.com/prepurchase2.html">More Essential Chord Progressions</a>&#8220;, or <a href="http://www.secretsofsongwriting.com/prepurchase2.html">“The Essential Secrets of Songwriting &#8211; Chord Progression Formulas.</a>”)</p>
<p>And as a final comment, I just want to clarify what I said at the beginning of this article. Many songwriters look for ways to make their progressions more interesting, when what&#8217;s really going on is that their progressions <span style="font-style:italic;">as is</span> just don&#8217;t really work. My advice is to make sure that your progressions in their basic form are working before you go looking for ways to make them interesting.</p>
<p>The analogy that works best here is: Don&#8217;t go looking for a really great new colour of paint for your house if it&#8217;s got structural problems. Fix the structure first, then look for paint.<br />
<a href="http://www.secretsofsongwriting.com/prepurchase2.html"><br />
</a><span style="font-weight:bold;"><a href="http://www.secretsofsongwriting.com/prepurchase2.html"><img style="width:100px;height:92px;float:left;border:0 solid initial;" src="http://www.secretsofsongwriting.com/images/all_6_sm.jpg" alt="6 Songwriting E-books" /></a></span><span style="color:red;"><a href="http://www.secretsofsongwriting.com/prepurchase2.html">&#8220;The Essential Secrets of Songwriting&#8221;</a> shows you how to write great songs. It&#8217;s just one of a suite of 6 songwriting e-books written by Gary Ewer. (His</span><span style="color:red;"> newest e-book, &#8221;The Essential Secrets of Songwriting- Chord Progression Formulas&#8221; is being offered for free when you purchase any other of his songwriting e-books.) <span style="font-weight:bold;">Let these six e-books show you </span><span style="font-style:italic;font-weight:bold;">every aspect</span><span style="font-weight:bold;"> of</span></span><span style="color:red;font-weight:bold;"> <span style="font-style:italic;">how</span></span><span style="font-weight:bold;"><span style="color:red;"><span style="font-style:italic;"> to write great songs!</span> </span></span><a href="http://www.secretsofsongwriting.com/prepurchase2.html"><span style="font-weight:bold;">Read more..</span></a></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Scary Yoga]]></title>
<link>http://greenspell.wordpress.com/2009/07/24/scary-yoga/</link>
<pubDate>Fri, 24 Jul 2009 03:17:04 +0000</pubDate>
<dc:creator>Yoga Witch</dc:creator>
<guid>http://greenspell.wordpress.com/2009/07/24/scary-yoga/</guid>
<description><![CDATA[Yoga teacher Kathryn Budig, who focuses on inversions and arm balances. Is it just me, or are invers]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><div id="attachment_1136" class="wp-caption alignleft" style="width: 310px"><img class="size-medium wp-image-1136" title="kbworkshop1008" src="http://greenspell.wordpress.com/files/2009/07/kbworkshop1008.jpg?w=300" alt="Yoga teacher Kathryn Budig, who focuses on inversions and arm balances." width="300" height="280" /><p class="wp-caption-text">Yoga teacher Kathryn Budig, who focuses on inversions and arm balances.</p></div>
<p>Is it just me, or are inversions scary? I love Plow Pose and Shoulder Stand, but Headstand and Handstand…hmmm. Scary. Let’s actually just stick with Headstand, because I am not physically ready for Handstand.</p>
<p><a href="http://www.youtube.com/watch?v=ubIAd-8FN50&#38;feature=related">Headstand</a> looks so easy. When my teacher demonstrates it in class, we students look at her and think: Heck – I can do that! But then we get into a kneeling position, set our forearms on the mat, lay the crown of our heads against the floor, and straighten our legs…suddenly, the idea of lift-off seems not only impossible, but scary! In fact, when the teacher came over to spot me once, I kicked up feebly, dropped down, then said, “I’m scared!” And of course, in that quiet room, <em>everyone </em>heard me. She laughed a little, and said, “It’s okay. I won’t let you go.”</p>
<p>I was less afraid after that, but still had images that she would hold on to my feet and the rest of me would fall over, and my neck would twist and I would fall on it, and… (You see – some poses invoke horrific fantasies in my head! Apparently, though, <a href="http://stanford.wellsphere.com/yoga-article/scary-yoga-and-lessons-from-zombies/461471">I’m not alone in this</a>.)</p>
<p>I did continue (for a while, anyway) to attempt to pull up into Headstand on my own, and for a while, I was not afraid of falling. It happened on my second try, and it wasn’t as bad as I thought it would be. But after a week or two of that…I got scared again, and didn’t want to keep falling over.</p>
<p>I think part of the problem is simply that I wasn’t yet strong enough. I’m much stronger now, a month later, but strong enough…? I don’t know. It’s important to have significant core strength for this pose.</p>
<p>But why are some poses so scary? It’s a funny thing – most people don’t associate yoga with fear. We think of it as relaxing, calming, centering. But true yoga asks us to <a href="http://brookshall.blogspot.com/2008/12/giving-up-fear.html">stretch ourselves farther than we think we can</a>. It challenges us, and asks us to take chances. Recently, I did a yoga DVD routine that featured many challenging poses, including doing Camel and the Breath of Fire at the same time. It scared me at first – I wasn’t sure I could do it. But the instructor said, “Be brave.” And I found it much easier to relax into it, and enjoy a sense of exhilaration that came with the pose.</p>
<p>I’d love to hear other people’s experiences with “scary” poses – Headstand, especially.</p>
<p><span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/n7img8taGWY&#038;rel=1&#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/n7img8taGWY&#038;rel=1&#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>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Graphs, Permutations, Characters, and Logspace]]></title>
<link>http://rjlipton.wordpress.com/2009/07/18/graphs-permutations-characters-and-logspace/</link>
<pubDate>Sat, 18 Jul 2009 19:01:02 +0000</pubDate>
<dc:creator>rjlipton</dc:creator>
<guid>http://rjlipton.wordpress.com/2009/07/18/graphs-permutations-characters-and-logspace/</guid>
<description><![CDATA[Counting the number of cycles of a permutation with connections to some classic open problems of com]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><span style="color:#0006cc;"><br />
<em>Counting the number of cycles of a permutation with connections to some classic open problems of complexity theory </em><br />
<span style="color:#000000;"> </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><img class="alignright size-full wp-image-3076" title="images" src="http://rjlipton.wordpress.com/files/2009/07/images8.jpeg" alt="images" width="99" height="107" /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Ken Regan is a great theorist, and probably the best chess player in the world who can really state what P=NP means. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Today I want to talk about a connection between logspace and the computation of the characters of a symmetric group. I also want to show how stupid I can be, which is not fun to point out. However, one of the agendas I had in starting this blog was to show the warts and all in research, and showing my errors is one way to do that. Oh well.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><!--more--></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Ken is a very strong chess player. Very. He is an International Master (IM), which is the rank below Grandmaster (GM). His international rating is 2405&#8212;ratings in the 2400&#8217;s are typical for IM&#8217;s, over 2500 for GM&#8217;s. I am impressed, when I played in high school my rating started with a 1 and the next digit was not large. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Ken has recently been working on some fascinating projects on detecting cheating in chess tournaments. The issue is simple: is a player using a computer program to help make the moves? As you probably know computer programs are terrific today. Top level programs are matched very well against the best chess players in the world. See for example <a href="http://en.wikipedia.org/wiki/Human-computer_chess_matches">this</a>. They are especially strong in tactical positions that routinely arise in real matches. So access to a computer would yield a great advantage to the player.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">During the 2006 Kramnik-Topalov world championship match cheating accusations against Kramnik were made. Ken was asked to try to decide how one might evaluate such accusations&#8212;that a player is making &#8220;too many&#8221; moves favored by a computer program, in this case the Fritz 9 program. Within days he posted his conclusion that Kramnik&#8217;s high coincidence to Fritz and another program, Rybka, was due to the unusually forcing nature of the games, Game 2 in particular. As Ken says, </span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;">&#8220;If you force play and limit your opponent&#8217;s options, don&#8217;t be surprised if he matches a computer&#8217;s moves!&#8221; </span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">See <a href="http://www.cse.buffalo.edu/~regan/chess/fidelity/">this</a> for further material and details on other cheating controversies and on ratings in general. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Ken deserves a thanks from all chess fans, at all levels, for his quick and insightful analysis of this controversy. Now let&#8217;s now turn to logspace and other concerns, and leave behind the coolness of high caliber chess.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> Cycle Languages </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">I need to define a number of languages that will be important for our discussion. All the languages consist of directed graphs with some additional property. The input will always be of the form: </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%28x_%7B1%7D%2Cy_%7B1%7D%29%2C%5Cdots%2C%28x_%7Bm%7D%2Cy_%7Bm%7D%29%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  (x_{1},y_{1}),\dots,(x_{m},y_{m})&amp;fg=000000' title='\displaystyle  (x_{1},y_{1}),\dots,(x_{m},y_{m})&amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"> where each <img src='http://l.wordpress.com/latex.php?latex=%7Bx_%7Bi%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x_{i}}&amp;fg=000000' title='{x_{i}}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7By_%7Bi%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{y_{i}}&amp;fg=000000' title='{y_{i}}&amp;fg=000000' class='latex' /> is in <img src='http://l.wordpress.com/latex.php?latex=%7B%5C%7B+1%2C%5Cdots%2Cn+%5C%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\{ 1,\dots,n \}}&amp;fg=000000' title='{\{ 1,\dots,n \}}&amp;fg=000000' class='latex' /> encoded as binary strings. The meaning of <img src='http://l.wordpress.com/latex.php?latex=%7B%28x%2Cy%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{(x,y)}&amp;fg=000000' title='{(x,y)}&amp;fg=000000' class='latex' /> is that there is a directed edge from <img src='http://l.wordpress.com/latex.php?latex=%7Bx%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{x}&amp;fg=000000' title='{x}&amp;fg=000000' class='latex' /> to <img src='http://l.wordpress.com/latex.php?latex=%7By%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{y}&amp;fg=000000' title='{y}&amp;fg=000000' class='latex' />. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">The first is the language <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Bcycle%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{cycle}}&amp;fg=000000' title='{L_{cycle}}&amp;fg=000000' class='latex' /> which consists of all inputs that encode a directed graph that consists of only vertex disjoint cycles. The next languages form a family of languages <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Bk%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{k}}&amp;fg=000000' title='{L_{k}}&amp;fg=000000' class='latex' /> and consist of all inputs that contain exactly <img src='http://l.wordpress.com/latex.php?latex=%7Bk%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{k}&amp;fg=000000' title='{k}&amp;fg=000000' class='latex' /> number of vertex disjoint directed cycles. Note, </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++L_%7Bk%7D+%5Csubsetneq+L_%7Bcycle%7D+%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  L_{k} \subsetneq L_{cycle} &amp;fg=000000' title='\displaystyle  L_{k} \subsetneq L_{cycle} &amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"> for any <img src='http://l.wordpress.com/latex.php?latex=%7Bk%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{k}&amp;fg=000000' title='{k}&amp;fg=000000' class='latex' />. Also </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++L_%7Bcycle%7D+%3D+L_%7B1%7D+%5Ccup+%5Cdots+%5Ccup+L_%7Bk%7D+%5Ccup+%5Cdots+%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  L_{cycle} = L_{1} \cup \dots \cup L_{k} \cup \dots &amp;fg=000000' title='\displaystyle  L_{cycle} = L_{1} \cup \dots \cup L_{k} \cup \dots &amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"> Finally, define <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Beven%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{even}}&amp;fg=000000' title='{L_{even}}&amp;fg=000000' class='latex' /> to be </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++L_%7B2%7D+%5Ccup+L_%7B4%7D+%5Ccup+%5Cdots+%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  L_{2} \cup L_{4} \cup \dots &amp;fg=000000' title='\displaystyle  L_{2} \cup L_{4} \cup \dots &amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> Basic Results </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Some definitions are in order. The class <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BAC%5E%7B0%7D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{AC^{0}}}&amp;fg=000000' title='{\mathsf{AC^{0}}}&amp;fg=000000' class='latex' /> is the uniform version of the famous class of constant depth and polynomial size circuits with AND/OR/NOT gates, and the AND/OR gates have unbounded fan-in. Also <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%7D%5Bm%5D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}}[m]}&amp;fg=000000' title='{\mathsf{ACC^{0}}[m]}&amp;fg=000000' class='latex' /> is the uniform class that also allows gates that compute mod <img src='http://l.wordpress.com/latex.php?latex=%7Bm%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{m}&amp;fg=000000' title='{m}&amp;fg=000000' class='latex' />. The class <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BL%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{L}}&amp;fg=000000' title='{\mathsf{L}}&amp;fg=000000' class='latex' /> denotes logspace. The following two theorems are straightforward. </span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Theorem: </strong> <em> The language <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Bcycle%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{cycle}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{cycle}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> is in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BAC%5E%7B0%7D%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{AC^{0}}}&amp;bg=e8e8e8&amp;fg=000000' title='{\mathsf{AC^{0}}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />. </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;"><em>Proof:</em> The key is that a directed graph has this form if and only if the in-degree and out-degree of each vertex is <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' />. Clearly, this can be checked as required. For example, the vertex <img src='http://l.wordpress.com/latex.php?latex=%7Bv%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{v}&amp;fg=000000' title='{v}&amp;fg=000000' class='latex' /> has in-degree at least <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' /> is equivalent to there is <em>some</em> <img src='http://l.wordpress.com/latex.php?latex=%7Bi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{i}&amp;fg=000000' title='{i}&amp;fg=000000' class='latex' /> so that <img src='http://l.wordpress.com/latex.php?latex=%7B%28x_%7Bi%7D%2Cy_%7Bi%7D%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{(x_{i},y_{i})}&amp;fg=000000' title='{(x_{i},y_{i})}&amp;fg=000000' class='latex' /> is an edge and <img src='http://l.wordpress.com/latex.php?latex=%7Bv+%3D+y_%7Bi%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{v = y_{i}}&amp;fg=000000' title='{v = y_{i}}&amp;fg=000000' class='latex' />. <img src='http://l.wordpress.com/latex.php?latex=%5CBox%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\Box&amp;fg=000000' title='\Box&amp;fg=000000' class='latex' /></span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Theorem: </strong> <em> For any <img src='http://l.wordpress.com/latex.php?latex=%7Bk+%5Cge+1%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{k \ge 1}&amp;bg=e8e8e8&amp;fg=000000' title='{k \ge 1}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />, the language <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Bk%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{k}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{k}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> is in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BL%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{L}}&amp;bg=e8e8e8&amp;fg=000000' title='{\mathsf{L}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />. </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> Logspace Complete Problems </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">A language is <em>logspace complete</em>, for us, always with respect to <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BAC%5E%7B0%7D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{AC^{0}}}&amp;fg=000000' title='{\mathsf{AC^{0}}}&amp;fg=000000' class='latex' /> reductions. The following theorem and proof is based on Stephen Cook&#8217;s and Pierre McKenzie&#8217;s classic paper&#8212;<a href="http://portal.acm.org/citation.cfm?id=32814">Problems complete for deterministic logarithmic space</a>. </span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Theorem: </strong> <em> It is <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BL%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{L}}&amp;bg=e8e8e8&amp;fg=000000' title='{\mathsf{L}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />-complete to tell <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7B1%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{1}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{1}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> from <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7B3%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{3}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{3}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />, or any <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Bk%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{k}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{k}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> from <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Bk%2B2%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{k+2}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{k+2}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />. Hence none of these tasks is in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5Bp%5D%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[p]}}&amp;bg=e8e8e8&amp;fg=000000' title='{\mathsf{ACC^{0}[p]}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />, for any prime <img src='http://l.wordpress.com/latex.php?latex=%7Bp%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{p}&amp;bg=e8e8e8&amp;fg=000000' title='{p}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />. </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">I have a proof of this theorem, but Ken worked out a better proof. The key parts of the proof were already in Cook-McKenzie, and Ken and Pierre worked out the rest. I thank them for allowing their proof to be <a href="http://rjlipton.wordpress.com/files/2009/07/logspace1.pdf">included here</a>. If you like you can skip the proof, but Ken worked hard to get all the details right. So please read it now. Or skip it now and come back later on. Or both.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> Even Cycle Theorem </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">So if it&#8217;s hard to tell <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' /> cycle from <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' />, how about <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' /> from <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' />, or <img src='http://l.wordpress.com/latex.php?latex=%7B2+%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2 }&amp;fg=000000' title='{2 }&amp;fg=000000' class='latex' />from <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' />? Long before I knew <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' /> from <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' /> is hard, I proved the following, which seems to be new as far as Ken and Pierre and some others can tell.</span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Theorem: </strong> <em> The language <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Beven%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{even}}&amp;bg=e8e8e8&amp;fg=000000' title='{L_{even}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> is in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B2%5D%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[2]}}&amp;bg=e8e8e8&amp;fg=000000' title='{\mathsf{ACC^{0}[2]}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />. </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">A corollary of this is that <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Beven%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{even}}&amp;fg=000000' title='{L_{even}}&amp;fg=000000' class='latex' /> cannot be <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BL%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{L}}&amp;fg=000000' title='{\mathsf{L}}&amp;fg=000000' class='latex' /> complete. This is an unconditional result, because otherwise it would violate known lower bounds on <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B2%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[2]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[2]}}&amp;fg=000000' class='latex' />. More on this in the next section.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Every input that describes a directed graph of disjoint cycles can be viewed as describing a permutation. This is the key to the proof of the theorem. Therefore, we need some basic facts about permutations. While they are simple and classic properties, they are surprisingly useful and powerful. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Each permutation has a value associated with it the number of inversions of the permutation, <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7Binv%7D%28%5Cpi%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{inv}(\pi)}&amp;fg=000000' title='{\mathsf{inv}(\pi)}&amp;fg=000000' class='latex' />. An <em>inversion</em> is a pair of indices <img src='http://l.wordpress.com/latex.php?latex=%7Bi%2Cj%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{i,j}&amp;fg=000000' title='{i,j}&amp;fg=000000' class='latex' /> so that <img src='http://l.wordpress.com/latex.php?latex=%7Bi%26%2360%3Bj%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{i&lt;j}&amp;fg=000000' title='{i&lt;j}&amp;fg=000000' class='latex' /> and yet <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi_%7Bi%7D+%26%2362%3B+%5Cpi_%7Bj%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi_{i} &gt; \pi_{j}}&amp;fg=000000' title='{\pi_{i} &gt; \pi_{j}}&amp;fg=000000' class='latex' />. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Moreover, the following classic lemma is the key:</span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Lemma: </strong> <em> For any permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;bg=e8e8e8&amp;fg=000000' title='{\pi}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> on <img src='http://l.wordpress.com/latex.php?latex=%7Bn%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{n}&amp;bg=e8e8e8&amp;fg=000000' title='{n}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> letters, </em></span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><em><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Ctext%7Bnumber+of+cycles+of+%7D+%5Cpi+%5Cequiv+n+%2B+%5Cmathsf%7Binv%7D%28%5Cpi%29+%5Cbmod+2.+%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \text{number of cycles of } \pi \equiv n + \mathsf{inv}(\pi) \bmod 2. &amp;bg=e8e8e8&amp;fg=000000' title='\displaystyle  \text{number of cycles of } \pi \equiv n + \mathsf{inv}(\pi) \bmod 2. &amp;bg=e8e8e8&amp;fg=000000' class='latex' /></em></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><em> </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">We now turn to the proof of the even cycle theorem (ECT):</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><em>Proof:</em> Recall that we could view the input graph as a permutation. We want to calculate the parity of the number of cycles. By the above lemma, it is enough to calculate the number of letters and the number of inversions of the permutation. Then, the theorem will follow by the lemma. But, computing each of these is quite simple for a constant depth polynomial circuit that has mod <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' /> gates. To compute the number of inversions one needs only try all pairs of <img src='http://l.wordpress.com/latex.php?latex=%7Bi%2Cj%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{i,j}&amp;fg=000000' title='{i,j}&amp;fg=000000' class='latex' /> and see if they form an inversion. If they do, count the number of them mod <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' />. This completes the proof. <img src='http://l.wordpress.com/latex.php?latex=%5CBox%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\Box&amp;fg=000000' title='\Box&amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> I Was Stupid </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Quite a while ago I began to look at directed graphs that consisted only of vertex disjoint cycles. I could encode certain problems into this simple class of graphs, and began to try to understand the complexity of telling the number of cycles and other properties. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">After a while I realized that they could be viewed as permutations in a natural way. This eventually led me to prove a theorem that was equivalent to the ECT. I then started to think about how to tell how many cycles a permutation had modulo <img src='http://l.wordpress.com/latex.php?latex=%7B4%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{4}&amp;fg=000000' title='{4}&amp;fg=000000' class='latex' />. This seemed, at the time, to be a reasonable possibility. I tried a wide range of tricks. They all were based on the fact that mod <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' /> gates can compute mod <img src='http://l.wordpress.com/latex.php?latex=%7B4%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{4}&amp;fg=000000' title='{4}&amp;fg=000000' class='latex' />. This is done by a kind of simple amplification trick. So I tried many graph and permutation constructions that attempted to reduce mod <img src='http://l.wordpress.com/latex.php?latex=%7B4%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{4}&amp;fg=000000' title='{4}&amp;fg=000000' class='latex' /> to mod <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' />.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">I had expected the ECT to generalize into statements like this:</span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%7B%5Ccup_k+L_%7B4k%7D+%5Cin+%5Cmathsf%7BACC%5E%7B0%7D%5B2%5D%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\cup_k L_{4k} \in \mathsf{ACC^{0}[2]}}&amp;bg=e8e8e8&amp;fg=000000' title='{\cup_k L_{4k} \in \mathsf{ACC^{0}[2]}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> </span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">or </span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%7B%5Ccup_k+L_%7B3k%7D+%5Cin+%5Cmathsf%7BACC%5E%7B0%7D%5B3%5D%7D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\cup_k L_{3k} \in \mathsf{ACC^{0}[3]}}&amp;bg=e8e8e8&amp;fg=000000' title='{\cup_k L_{3k} \in \mathsf{ACC^{0}[3]}}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />. </span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">My hope was all in vain, since each of these is impossible. When I say impossible, I mean that these statements are actually <em>false</em>&#8212;not just if-horses-can-fly-then-pigs-can-whistle unlikely, but really false. The reason is the brilliant work of Roman Smolensky. I am a bit embarrassed that I took a while&#8212;make that a long while&#8212;to see the connection with his well known work. I can only say that I guess I got caught in the trees, and could not see the forest. In any event I now understand now what is going on much better.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Recall Smolensky proved that <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5Bm%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[m]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[m]}}&amp;fg=000000' class='latex' /> is not in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5Bp%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[p]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[p]}}&amp;fg=000000' class='latex' /> when <img src='http://l.wordpress.com/latex.php?latex=%7Bp%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{p}&amp;fg=000000' title='{p}&amp;fg=000000' class='latex' /> is a prime and <img src='http://l.wordpress.com/latex.php?latex=%7Bm%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{m}&amp;fg=000000' title='{m}&amp;fg=000000' class='latex' /> is not a power of <img src='http://l.wordpress.com/latex.php?latex=%7Bp%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{p}&amp;fg=000000' title='{p}&amp;fg=000000' class='latex' />. So <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B2%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[2]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[2]}}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B3%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' class='latex' /> are not in each other, and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B6%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' class='latex' /> is not in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B3%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' class='latex' /> or in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B5%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[5]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[5]}}&amp;fg=000000' class='latex' />. This is not symmetrical, though&#8212;e.g. it does not give you that <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B5%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[5]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[5]}}&amp;fg=000000' class='latex' /> is not in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B6%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' class='latex' />. Indeed, <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B6%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' class='latex' /> is a class that gets more annoying every year. It <em>should</em> be small, but we currently cannot even tell that it&#8217;s different from NP. Why do we think we can separate P from NP when we cannot even separate it from <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B6%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' class='latex' />? Thus counting modulo primes versus composites is the frontier of lower bounds, and has been for over two decades. I thought counting cycles would behave like counting 1s in a binary string, but be more important because of how graphs relate to Turing machines.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">It should now be clear why <img src='http://l.wordpress.com/latex.php?latex=%7BX%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{X}&amp;fg=000000' title='{X}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7BY%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{Y}&amp;fg=000000' title='{Y}&amp;fg=000000' class='latex' /> are impossible. Look at <img src='http://l.wordpress.com/latex.php?latex=%7BX%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{X}&amp;fg=000000' title='{X}&amp;fg=000000' class='latex' />&#8212;if it were true, then you could tell <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' /> cycles apart from <img src='http://l.wordpress.com/latex.php?latex=%7B4%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{4}&amp;fg=000000' title='{4}&amp;fg=000000' class='latex' /> cycles. By Theorem that would imply <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B2%5D%7D+%3D+%5Cmathsf%7BL%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[2]} = \mathsf{L}}&amp;fg=000000' title='{\mathsf{ACC^{0}[2]} = \mathsf{L}}&amp;fg=000000' class='latex' />, which would contradict his famous result. Likewise <img src='http://l.wordpress.com/latex.php?latex=%7BY%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{Y}&amp;fg=000000' title='{Y}&amp;fg=000000' class='latex' /> would put telling <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' /> from <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' /> cycles into <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B3%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' class='latex' />, which is impossible.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Further, if I promise that the input graph <img src='http://l.wordpress.com/latex.php?latex=%7BG%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{G}&amp;fg=000000' title='{G}&amp;fg=000000' class='latex' /> is either a single cycle, or is <img src='http://l.wordpress.com/latex.php?latex=%7B3%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{3}&amp;fg=000000' title='{3}&amp;fg=000000' class='latex' /> disjoint cycles, with <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B3%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[3]}}&amp;fg=000000' class='latex' /> circuits, can you tell apart these two cases? No. Likewise if I promise <img src='http://l.wordpress.com/latex.php?latex=%7BG%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{G}&amp;fg=000000' title='{G}&amp;fg=000000' class='latex' /> forms either <img src='http://l.wordpress.com/latex.php?latex=%7B2%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{2}&amp;fg=000000' title='{2}&amp;fg=000000' class='latex' /> cycles or <img src='http://l.wordpress.com/latex.php?latex=%7B4%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{4}&amp;fg=000000' title='{4}&amp;fg=000000' class='latex' />, or <img src='http://l.wordpress.com/latex.php?latex=%7B9%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{9}&amp;fg=000000' title='{9}&amp;fg=000000' class='latex' /> versus <img src='http://l.wordpress.com/latex.php?latex=%7B11%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{11}&amp;fg=000000' title='{11}&amp;fg=000000' class='latex' /> cycles, the answer is no. However, you can tell <img src='http://l.wordpress.com/latex.php?latex=%7B9%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{9}&amp;fg=000000' title='{9}&amp;fg=000000' class='latex' /> versus <img src='http://l.wordpress.com/latex.php?latex=%7B10%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{10}&amp;fg=000000' title='{10}&amp;fg=000000' class='latex' /> by ECT.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> Characters </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">There is a way to try to generalize the ECT based on <a href="http://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group">character theory</a> for the symmetric group. This attempt is an approach to prove a theorem like </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cmathsf%7BACC%5E%7B0%7D%7D%5B6%5D+%3D+%5Cmathsf%7BL%7D.%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \mathsf{ACC^{0}}[6] = \mathsf{L}.&amp;fg=000000' title='\displaystyle  \mathsf{ACC^{0}}[6] = \mathsf{L}.&amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"> Or if you think this is false, this approach can be viewed as <a href="http://rjlipton.wordpress.com/2009/05/30/papadimitriou-and-rct/">RCT</a>: reverse complexity theory. What we may be able to prove is that characters of the symmetric group are &#8220;hard&#8221; to compute. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">For us characters will just be mappings from the symmetric group to the integers. A character <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cchi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\chi}&amp;fg=000000' title='{\chi}&amp;fg=000000' class='latex' /> has many properties, but one of the key properties is that if <img src='http://l.wordpress.com/latex.php?latex=%7B%5Calpha%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\alpha}&amp;fg=000000' title='{\alpha}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbeta%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\beta}&amp;fg=000000' title='{\beta}&amp;fg=000000' class='latex' /> are conjugate permutations, then </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cchi%28%5Calpha%29+%3D+%5Cchi%28%5Cbeta%29.%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \chi(\alpha) = \chi(\beta).&amp;fg=000000' title='\displaystyle  \chi(\alpha) = \chi(\beta).&amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"> Recall two permutations are conjugate if they have the same cyclic structure. Functions that are constant on all permutations with the same cyclic structure are called <em>class functions</em>, since they are constant on a conjugacy class. The characters form a linear basis of the class functions.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Consider the problem of telling apart two permutation types: <img src='http://l.wordpress.com/latex.php?latex=%7B%5Calpha%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\alpha}&amp;fg=000000' title='{\alpha}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbeta%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\beta}&amp;fg=000000' title='{\beta}&amp;fg=000000' class='latex' />. Suppose that <img src='http://l.wordpress.com/latex.php?latex=%7B%5Calpha%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\alpha}&amp;fg=000000' title='{\alpha}&amp;fg=000000' class='latex' /> has one cycle, and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbeta%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\beta}&amp;fg=000000' title='{\beta}&amp;fg=000000' class='latex' /> has three cycles of equal size. The class <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B2%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[2]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[2]}}&amp;fg=000000' class='latex' /> cannot tell these apart, for that would again yield an impossible collapse. However, can <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5B6%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[6]}}&amp;fg=000000' class='latex' />, for example, tell them apart? This is the main open question that we are interested in. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Since the characters form a linear basis of the class functions, if <img src='http://l.wordpress.com/latex.php?latex=%7B%5Calpha%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\alpha}&amp;fg=000000' title='{\alpha}&amp;fg=000000' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbeta%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\beta}&amp;fg=000000' title='{\beta}&amp;fg=000000' class='latex' /> are not conjugate permutations, there is a character <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cchi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\chi}&amp;fg=000000' title='{\chi}&amp;fg=000000' class='latex' />, so that </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cchi%28%5Calpha%29+%5Cneq+%5Cchi%28%5Cbeta%29.%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \chi(\alpha) \neq \chi(\beta).&amp;fg=000000' title='\displaystyle  \chi(\alpha) \neq \chi(\beta).&amp;fg=000000' class='latex' /></span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Theorem: </strong> <em> Suppose that for <img src='http://l.wordpress.com/latex.php?latex=%7Bn%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{n}&amp;bg=e8e8e8&amp;fg=000000' title='{n}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />, all the characters of the symmetric group on <img src='http://l.wordpress.com/latex.php?latex=%7Bn%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{n}&amp;bg=e8e8e8&amp;fg=000000' title='{n}&amp;bg=e8e8e8&amp;fg=000000' class='latex' /> letters can be computed by <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%7D%5Bm%5D%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}}[m]}&amp;bg=e8e8e8&amp;fg=000000' title='{\mathsf{ACC^{0}}[m]}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />, then </em></span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><em><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cmathsf%7BACC%5E%7B0%7D%7D%5Bm%5D+%3D+%5Cmathsf%7BL%7D.%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \mathsf{ACC^{0}}[m] = \mathsf{L}.&amp;bg=e8e8e8&amp;fg=000000' title='\displaystyle  \mathsf{ACC^{0}}[m] = \mathsf{L}.&amp;bg=e8e8e8&amp;fg=000000' class='latex' /></em></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><em> </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">I think that this theorem can be greatly strengthened, by carefully calculating which characters are actually needed&#8212;needed to tell one cycle from three cycles. Finally, note that one of the characters is always easy to compute: </span></span></p>
<blockquote><p><span style="color:#0006cc;"><span style="color:#000000;"><strong>Lemma: </strong> <em> For any permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;bg=e8e8e8&amp;fg=000000' title='{\pi}&amp;bg=e8e8e8&amp;fg=000000' class='latex' />, </em></span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><em><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cchi_%7B%5Ctext%7Bparity%7D%7D%28%5Cpi%29+%5Cequiv+%5Cmathsf%7Binv%7D%28%5Cpi%29+%5Cbmod+2+%26%2338%3Bbg%3De8e8e8%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \chi_{\text{parity}}(\pi) \equiv \mathsf{inv}(\pi) \bmod 2 &amp;bg=e8e8e8&amp;fg=000000' title='\displaystyle  \chi_{\text{parity}}(\pi) \equiv \mathsf{inv}(\pi) \bmod 2 &amp;bg=e8e8e8&amp;fg=000000' class='latex' /></em></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><em> </em></span></span></p></blockquote>
<p><span style="color:#0006cc;"><span style="color:#000000;">The character <img src='http://l.wordpress.com/latex.php?latex=%7B+%5Cchi_%7B%5Ctext%7Bparity%7D%7D+%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{ \chi_{\text{parity}} }&amp;fg=000000' title='{ \chi_{\text{parity}} }&amp;fg=000000' class='latex' /> is defined as follows. Every permutation can be written as the product of transpositions. Recall a transposition leaves all elements fixed, but switches two elements. A permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' /> can be written in many ways as the product of transpositions, but the parity of the number of transpositions is always the same. Thus, we can assign to <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' /> a value <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cchi_%7B%5Ctext%7Bparity%7D%7D%28%5Cpi%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\chi_{\text{parity}}(\pi)}&amp;fg=000000' title='{\chi_{\text{parity}}(\pi)}&amp;fg=000000' class='latex' /> in <img src='http://l.wordpress.com/latex.php?latex=%7B%5C%7B+-1%2C+%2B1+%5C%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\{ -1, +1 \}}&amp;fg=000000' title='{\{ -1, +1 \}}&amp;fg=000000' class='latex' />: the value is <img src='http://l.wordpress.com/latex.php?latex=%7B-1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{-1}&amp;fg=000000' title='{-1}&amp;fg=000000' class='latex' /> for odd permutations and is <img src='http://l.wordpress.com/latex.php?latex=%7B%2B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{+1}&amp;fg=000000' title='{+1}&amp;fg=000000' class='latex' /> for even permutations. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">Another way to view this lemma is that there is a summation that computes <img src='http://l.wordpress.com/latex.php?latex=%7B+%5Cchi_%7B%5Ctext%7Bparity%7D%7D+%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{ \chi_{\text{parity}} }&amp;fg=000000' title='{ \chi_{\text{parity}} }&amp;fg=000000' class='latex' />. For any permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' />, </span></span></p>
<p align="center"><span style="color:#0006cc;"><span style="color:#000000;"><img src='http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle++%5Cchi_%7B%5Ctext%7Bparity%7D%7D%28%5Cpi%29+%5Cequiv+%5Csum_%7Bi%26%2360%3Bj%7D+%5C%2C+%5Culcorner+%5Cpi_%7Bi%7D+%26%2362%3B+%5Cpi_%7Bj%7D+%5Curcorner+%5Cbmod+2.%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\displaystyle  \chi_{\text{parity}}(\pi) \equiv \sum_{i&lt;j} \, \ulcorner \pi_{i} &gt; \pi_{j} \urcorner \bmod 2.&amp;fg=000000' title='\displaystyle  \chi_{\text{parity}}(\pi) \equiv \sum_{i&lt;j} \, \ulcorner \pi_{i} &gt; \pi_{j} \urcorner \bmod 2.&amp;fg=000000' class='latex' /></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"> Here <img src='http://l.wordpress.com/latex.php?latex=%7B%5Culcorner+X+%5Curcorner%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\ulcorner X \urcorner}&amp;fg=000000' title='{\ulcorner X \urcorner}&amp;fg=000000' class='latex' /> is <img src='http://l.wordpress.com/latex.php?latex=%7B1%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{1}&amp;fg=000000' title='{1}&amp;fg=000000' class='latex' /> if <img src='http://l.wordpress.com/latex.php?latex=%7BX%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{X}&amp;fg=000000' title='{X}&amp;fg=000000' class='latex' /> is true, and <img src='http://l.wordpress.com/latex.php?latex=%7B0%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{0}&amp;fg=000000' title='{0}&amp;fg=000000' class='latex' />, otherwise. The question, is are there similar summations for any other interesting characters?</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">There are other characters that are also easy to compute, but they seem to be uninteresting from a complexity point of view. For example, let <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cchi_%7B%5Ctext%7Bfixed%7D%7D%28%5Cpi%29%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\chi_{\text{fixed}}(\pi)}&amp;fg=000000' title='{\chi_{\text{fixed}}(\pi)}&amp;fg=000000' class='latex' /> be the number of points that the permutation <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cpi%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\pi}&amp;fg=000000' title='{\pi}&amp;fg=000000' class='latex' /> leaves fixed. This is easy to compute too, but does not seem to help.</span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;"><strong> Open Problems </strong></span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">There are many interesting open questions. First, are there any applications of the ECT? Second, are there some characters that have special formulas like <img src='http://l.wordpress.com/latex.php?latex=%7B+%5Cchi_%7B%5Ctext%7Bparity%7D%7D+%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{ \chi_{\text{parity}} }&amp;fg=000000' title='{ \chi_{\text{parity}} }&amp;fg=000000' class='latex' /> does? I also do not know if <img src='http://l.wordpress.com/latex.php?latex=%7BL_%7Beven%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{L_{even}}&amp;fg=000000' title='{L_{even}}&amp;fg=000000' class='latex' /> is in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%5Bp%5D%7D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}[p]}}&amp;fg=000000' title='{\mathsf{ACC^{0}[p]}}&amp;fg=000000' class='latex' /> for any odd prime <img src='http://l.wordpress.com/latex.php?latex=%7Bp%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{p}&amp;fg=000000' title='{p}&amp;fg=000000' class='latex' />. </span></span></p>
<p><span style="color:#0006cc;"><span style="color:#000000;">The key question is what can we say about the computational cost of computing characters in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%7D%5Bm%5D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}}[m]}&amp;fg=000000' title='{\mathsf{ACC^{0}}[m]}&amp;fg=000000' class='latex' /> for <img src='http://l.wordpress.com/latex.php?latex=%7Bm%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{m}&amp;fg=000000' title='{m}&amp;fg=000000' class='latex' /> composite? If the answer is that <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cmathsf%7BACC%5E%7B0%7D%7D%5Bm%5D%7D%26%2338%3Bfg%3D000000&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\mathsf{ACC^{0}}[m]}&amp;fg=000000' title='{\mathsf{ACC^{0}}[m]}&amp;fg=000000' class='latex' /> can compute all characters, then as we observed, that would solve a long standing open problem. </span></span></p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Exhileration!]]></title>
<link>http://perusals.wordpress.com/2009/07/04/exhileration/</link>
<pubDate>Sat, 04 Jul 2009 15:43:59 +0000</pubDate>
<dc:creator>Abby</dc:creator>
<guid>http://perusals.wordpress.com/2009/07/04/exhileration/</guid>
<description><![CDATA[tripod headstand Headstand = success!! With not a rain drop in sight and a morning free, I thought t]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><div id="attachment_483" class="wp-caption alignright" style="width: 310px"><img class="size-medium wp-image-483" title="tripod headstand" src="http://perusals.wordpress.com/files/2009/07/cc08a684f79e0194_tripod.jpg?w=300" alt="tripod headstand" width="300" height="231" /><p class="wp-caption-text">tripod headstand</p></div>
<p>Headstand = success!!</p>
<p>With not a rain drop in sight and a morning free, I thought to myself, &#8220;What  a perfect day for an outside yoga practice!&#8221;  So I fished an old bath towel out of the laundry bin and spread it out under a big oak tree in Ed&#8217;s yard.  What a beautiful way to practice.  Physically connecting to the earth made such a difference &#8212; especially in being conscious of my core since the real earth feels nothing like a yoga studio floor ::wink::</p>
<p>When I got into wide-legged forward bend, I figured it would be a good time to work on my tripod.  I&#8217;ve <a href="http://perusals.wordpress.com/2009/04/04/yoga-school-day-13/">had issues</a> with inversion in the past, but I was feeling pretty safe and strong today. Setting up head and hands, I  scooted my knees onto my triceps and hung out there for a few breaths. Already feeling pretty pleased with myself, I&#8217;m not sure what inspired me to try the full on headstand, but I took a deep breath and slowly brought my legs overhead &#8230; AND STAYED THERE.  OMG!!  I WAS IN A HEADSTAND!  And not a panic attack or anxiety in sight!  Literally.  All I could see was the green grass tickling the upside down blue sky.  I didn&#8217;t stay up for  very long and made a resounding THUD when I landed.  But holy cow kids, I did it!  I had no idea I had it in me.  I figured I didn&#8217;t have the strength to get my legs up over my head, but you know what?  I DO!</p>
<p>Right now I feel on top of the world &#8212; strong and capable and so grateful for my body for being strong and my mind for not freaking the eff out like it usually does.</p>
<p>A fellow yogi recently left a Facebook status update saying that headstands are a great way to start the day.  I&#8217;m thinking she&#8217;s on to something big. <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_smile.gif' alt=':)' class='wp-smiley' /> </p>
<p>Happy Independence Day!</p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Pincha Mayurasana: Forearm Stand]]></title>
<link>http://kylemilleryoga.com/2009/06/30/pincha-mayurasana-forearm-stand/</link>
<pubDate>Wed, 01 Jul 2009 01:35:13 +0000</pubDate>
<dc:creator>Kyle</dc:creator>
<guid>http://kylemilleryoga.com/2009/06/30/pincha-mayurasana-forearm-stand/</guid>
<description><![CDATA[the most beautiful interpretation. yoga in the park.                       Pincha Mayurasana is one ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><div class="mceTemp mceIEcenter">
<div style="text-align:left;">
<div id="attachment_881" class="wp-caption alignleft" style="width: 310px"><img class="size-medium wp-image-881" title="BKHASGDLArtofYoga400" src="http://kylemilleryoga.wordpress.com/files/2009/06/bkhasgdlartofyoga4001.jpg?w=300" alt="the most beautiful interpretation of forearm stand with a partner." width="300" height="300" /><p class="wp-caption-text">the most beautiful interpretation.</p></div>
<div class="mceTemp">
<div id="attachment_896" class="wp-caption alignright" style="width: 235px"><img class="size-full wp-image-896" title="sc002df8eb-1" src="http://kylemilleryoga.wordpress.com/files/2009/06/sc002df8eb-1.jpg" alt="yoga in the park." width="225" height="299" /><p class="wp-caption-text">yoga in the park.</p></div>
</div>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p>Pincha Mayurasana is one of my favorite poses.  Once you can find the balance, you body lifts right up and is completely, effortlessly supported by your forearms.  Doing this with a partner is a beautiful interpretation of the pose.  I first saw this done on the cover of <span style="text-decoration:underline;"><a href="http://www.amazon.com/Art-Yoga-Sharon-Gannon/dp/1584792078">The Art of Yoga</a></span> by Sharon and David. Then I saw my friends <a href="http://kylemilleryoga.com/2009/04/28/spiritual-food-kirtan/">Nahdi</a> and <a href="http://www.jivamuktiyoga.com/fms/teach_fm.html">Luis</a>, 2 amazing teachers from <a href="http://www.jivamuktiyoga.com/fms/index.html">Jivamukti</a>, do the pose.  Right now it is on a flyer for Jivamukti Yoga in the Park, featuring <a href="http://www.jivamuktiyoga.com/fms/teach_fm.html">Charlotte</a> and another teacher I&#8217;m afraid to say I don&#8217;t know. Regardless, they look amazing.  </div>
</div>
<p>I remember when I thought getting into forearm stand would be impossible.  Thanks to my teachers, I can lift up and hold the pose comfortably now.  I have even been able to help my friends and students get into this pose, which is truly rewarding.  If you want to be able to get in this pose, or at least give it a shot I recommend taking a class with <a href="http://www.sandyoga.com/">Sandhi</a>. She is an amazing acro-yogi and gives the best instructions for getting into inversions (she&#8217;s teaching an <a href="http://www.jivamuktiyoga.com/event/moreInfo.jsp?detailID=511">inversion immersion</a> at Jivamukti on July 19th).  If you want to give it a try yourself come into <a href="http://www.yogajournal.com/poses/2462">dolphin pose</a> (downward dog on your forearms) making sure that you wrists, elbows, and shoulders line up.  Look forward and pull your shoulders back in space so they are directly above your elbows.  Slowly walk your feet towards your hands.  When your hips are very high, lift one leg up.  Bend your standing leg, lowering the raised leg. Then straighten the standing leg extending the other leg back into the air.  Do this a couple times creating some momentum.  Kick off the standing leg, catching some air.  Try it again until you can hold yourself up.  You should really try this for the first time with a teacher or else you could hurt yourself.  Just remember that it is possible.</p>
<div id="attachment_874" class="wp-caption aligncenter" style="width: 610px"><img class="size-full wp-image-874" title="3026_1144756092006_1020617434_430830_57678_n" src="http://kylemilleryoga.wordpress.com/files/2009/06/3026_1144756092006_1020617434_430830_57678_n.jpg" alt="Nadhi and Luis!" width="600" height="400" /><p class="wp-caption-text">Nadhi and Luis!</p></div>
<div id="attachment_895" class="wp-caption aligncenter" style="width: 463px"><img class="size-full wp-image-895" title="4543_790541955429_809101_46172883_1313351_n" src="http://kylemilleryoga.wordpress.com/files/2009/06/4543_790541955429_809101_46172883_1313351_n1.jpg" alt="me and a friend.. it was our first try!!!!  we'll get better!" width="453" height="604" /><p class="wp-caption-text">me and a friend.. it was our first try!!!!  we&#39;ll get better!</p></div>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Upside down]]></title>
<link>http://zenfreckle.wordpress.com/2009/06/26/upside-down/</link>
<pubDate>Fri, 26 Jun 2009 16:58:10 +0000</pubDate>
<dc:creator>zenfreckle</dc:creator>
<guid>http://zenfreckle.wordpress.com/2009/06/26/upside-down/</guid>
<description><![CDATA[My roommate and I have rearranged our living space and I now have a really great space for my practi]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>My roommate and I have rearranged our living space and I now have a really great space for my practice &#8211; pictures to come soon! This has made it possible for me to practice inversions safely without having to move all the furniture around. </p>
<p>Inversions were a scary part of the practice for a very long time. I love doing supported headstand, shoulderstand, etc but doing them sans props&#8230;yikes! I guess I was always afraid that I couldn&#8217;t hold myself up. Slowly I have come to realize that I have the foundations of the poses and really can hold myself up. The trick now is to soften into the pose. To hold myself up without holding tension in my body. That will probably take years!</p>
<p>What poses do you find the most scary and how do you overcome them?  </p>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Les obres del Pla Zapatero obren una via a les empreses locals]]></title>
<link>http://rogergarcia.wordpress.com/2009/05/23/les-obres-del-pla-zapatero-obren-una-via-a-les-empreses-locals/</link>
<pubDate>Sat, 23 May 2009 19:09:13 +0000</pubDate>
<dc:creator>rogergarcia</dc:creator>
<guid>http://rogergarcia.wordpress.com/2009/05/23/les-obres-del-pla-zapatero-obren-una-via-a-les-empreses-locals/</guid>
<description><![CDATA[Les obres del Fons Estatal d’Inversió Local (FEIL) van a molt bon ritme i podem preveure que l’Ajunt]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img src="http://rogergarcia.wordpress.com/files/2009/05/abat-escarre.jpg?w=150" alt="Abat escarre" title="Abat escarre" width="150" height="120" class="alignleft size-thumbnail wp-image-68" />
<div style="text-align:justify;">Les obres del <a href="http://www.ajrubi.es/ajrubi/apartats/index.php?apartat=2106">Fons Estatal d’Inversió Local (FEIL)</a> van a molt bon ritme i podem preveure que l’Ajuntament complirà els seus compromisos. Es faran totes les 31 obres previstes i es faran dins del termini que ens marca el Pla Zapatero. Les empreses estan cobrant puntualment a 30 dies. S’estan contractant els treballadors que les empreses havien compromès, molts d’ells procedents de l’atur. Aquestes obres suposaran 12,5 milions d’euros invertits a Rubí per millorar la ciutat. I les molèsties als veïns s’han reduït al mínim.<br />
Però hi un aspecte que caldrà millorar. La participació de les empreses locals en les obres de la ciutat.<br />
Hi ha poca tradició en aquest sentit i les nostres empreses tenen poca experiència en processos de licitació ja fins ara havien escollit l’opció de la promoció privada i pràcticament no es presentaven a les licitacions municipals, les quals tenen uns marges més estrets i uns condicionants administratius estrictes.<br />
Ara be, la crisi del mon de la construcció privada fa que les empreses s’hagin plantejat la participació en les obres públiques i des de l’Ajuntament hem fet un seguit d’accions per afavorir-la.<br />
Com a resultat, les empreses amb seu a Rubí han presentat pressupost per fer 26 de les 31 obres finançades pel Pla Zapatero, i han estat contractades a 7 d’aquestes obres. Dit d’una altra forma, en 3 de cada 10 obres locals a les que s’han presentat han aconseguit el contracte.<br />
Aquest percentatge és encara superior si ens fixem en els contractes que s’han endut les empreses locals en aquelles obres que no requerien la condició de classificació perquè tenen un pressupost inferior als 350.000 €. Un total de 12 empreses de Rubí s’han presentat a les 15 obres que compleixen aquestes característiques i 6 s’han endut el contracte: el 50 %.<br />
Per altra part, vam demanar a les grans empreses que han guanyat els concursos de les grans obres, que subcontractessin empreses de Rubí i que els treballadors aturats els agafessin de les borses de treball locals. Fruit d’aquesta demanda municipal s’ha subcontractat empreses de Rubí en 6 obres i 130 persones de les borses locals hauran sortit de l’atur.<br />
Resumint: en 13 obres de les 31 obres del Pla Zapatero a Rubí participen empreses locals de forma directa o indirecta i, 770 persones treballaran en el conjunt dels projectes.<br />
Malgrat ser un balanç positiu, l’Ajuntament pren nota d’allò que cal millorar i ja hem començat a treballar per facilitar encara més l’accés d’empreses locals tant a les obres municipals d’aquest any com als projectes que s’inclouran en el segon Pla Zapatero que es prepara per a l’any vinent. Per exemple ja hem dividit els nous projectes d’inversió ordinària en paquets més petits, (amb un esforç, en temps i recursos municipals, important.) de manera que es poden presentar els nostres autònoms i les petites i mitjanes empreses locals sense necessitat de disposar de la classificació. A més, facilitarem la màxima informació i transparència des de l’Ajuntament i anirem introduint tots els elements que ens permeti el marc legal, per procurar que les nostres empreses puguin ser adjudicatàries de les nostres obres i també les de la comarca. I això ho volem fer colze a colze amb els agents econòmics locals
</div>
</div>]]></content:encoded>
</item>
<item>
<title><![CDATA[Obres Pla Zapatero a tot gas!]]></title>
<link>http://rogergarcia.wordpress.com/2009/05/10/obres-pla-zapatero-a-tot-gas/</link>
<pubDate>Sun, 10 May 2009 11:28:29 +0000</pubDate>
<dc:creator>rogergarcia</dc:creator>
<guid>http://rogergarcia.wordpress.com/2009/05/10/obres-pla-zapatero-a-tot-gas/</guid>
<description><![CDATA[Estem contents. L&#8217;arrencada ha estat bona. Totes les obres s&#8217;han iniciat a la data previ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img src="http://rogergarcia.wordpress.com/files/2009/05/obres.jpg" alt="obres" title="obres" width="118" height="103" class="alignleft size-full wp-image-62" />
<div style="text-align:justify;">Estem contents. L&#8217;arrencada ha estat bona. Totes les obres s&#8217;han iniciat a la data prevista i a un molt bon ritme.<br />
Heu passat per davant de l&#8217;Ajuntament? Ja no hi és l&#8217;edifici que estava en mal estat. Aviat veurem l&#8217;inici d&#8217;un nou edificis d&#8217;oficines municipals amb un disseny i una funcionalitat al servei dels ciutadans.<br />
I l&#8217;antiga fàbrica de Can Sedó a Ca n&#8217;Alzamora? Ja no en queda res. Aquest serà l&#8217;espai que ocuparà en un futur una nova escola bressol. Un espai que queda entre el CEIP Ca n&#8217;Alzamora i el parc que s&#8217;està remodelant i posant al dia.<br />
Heu patit les cues quan veniu de Sant Quirze o a la sortida de l&#8217;Autopista a l&#8217;alçada del Pinar? Deu ni do les retencions dels primers dies! La causa: les obres del Vial entre rotondes que unirà les dues carreteres, (la BP-1503 i la C-1413a al Pol·ligon industrial de la Bastida) tindrà quatre carrils i voreres de quatre metres amb bancs, enllumenat i arbres. Quin canvi notarem! Com millorarà la circulació de vehicles i de vianants!<br />
Son veïns del carrer Abat Escarré o de la zona? Quants inconvenients! Les obres duraran el mínim imprescindible, però la previssió és d&#8217;uns cinc mesos. Ara be per reduir les molèsties les obres es fan per trams i deixant sempre lliure una de les voreres. Els inconvenients hi son però quan acabin les obres tindrem un carrer nou, on es mantindrà l&#8217;aparcament a les dues bandes al primer tram (Progrés &#8211; Monturiol) deixant voreres de metre i mig i al segon tram serà amb una zona d&#8217;aparcament, però de prioritat pels vianants amb arbres i bancs.<br />
I el parc de la bòbila Saltó? L&#8217;obra més cara i complicada de les que estem fent. Un parc amb 8.000 m2 de superfície i amb un desnivell màxim de vuit metres. Quants camions que entraran i sortiran pels carrers de Santa Fe i del General Castaños! Quin enrenou! Però quin nou espai verd tindrem a la zona Nord de la ciutat l&#8217;any vinent!<br />
I les escoles? Elles saben el que es conviure amb obres i cal agrair la seva col·laboració i comprensió.<br />
I podríem seguir amb la resta de les 31 obres del FEIL, del Pla Zapatero.<br />
Totes funcionen a bon ritme i de moment sense entrebancs. Les empreses fan la seva feina, cobren puntualment cada mes tal i com està previst i donen feina a moltes persones.<br />
Hem de demanar disculpes per les molèsties, encara que sabent que moltes de les obres acabaran a l&#8217;estiu i la resta abans de finals d&#8217;any.
</div>
</div>]]></content:encoded>
</item>

</channel>
</rss>
