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<channel>
	<title>spectral &amp;laquo; WordPress.com Tag Feed</title>
	<link>http://en.wordpress.com/tag/spectral/</link>
	<description>Feed of posts on WordPress.com tagged "spectral"</description>
	<pubDate>Fri, 04 Dec 2009 21:47:52 +0000</pubDate>

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

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<title><![CDATA[28.11.2009 - Autumn From Hell im Schützenhaus Hohenstein-Ernstthal]]></title>
<link>http://xangelofdark.wordpress.com/2009/11/29/28-11-2009-autumn-from-hell-im-schutzenhaus-hohenstein-ernstthal/</link>
<pubDate>Sun, 29 Nov 2009 14:30:54 +0000</pubDate>
<dc:creator>AngelOfDark</dc:creator>
<guid>http://xangelofdark.wordpress.com/2009/11/29/28-11-2009-autumn-from-hell-im-schutzenhaus-hohenstein-ernstthal/</guid>
<description><![CDATA[Zum Samstagabend bißchen Metal? Bin ich doch glatt dabei &#8211; also in den Zug geschwungen nach Ho]]></description>
<content:encoded><![CDATA[Zum Samstagabend bißchen Metal? Bin ich doch glatt dabei &#8211; also in den Zug geschwungen nach Ho]]></content:encoded>
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<title><![CDATA[Iannis Xenakis - Metastasis (Spectral View)]]></title>
<link>http://reaktorplayer.wordpress.com/2009/11/13/1350/</link>
<pubDate>Fri, 13 Nov 2009 15:57:53 +0000</pubDate>
<dc:creator>reaktorplayer</dc:creator>
<guid>http://reaktorplayer.wordpress.com/2009/11/13/1350/</guid>
<description><![CDATA[Iannis Xenakis &#8211; Metastasis (Spectral View) From Wiki: Iannis Xenakis (Ιωάννης Ιάννης Ξενάκης)]]></description>
<content:encoded><![CDATA[Iannis Xenakis &#8211; Metastasis (Spectral View) From Wiki: Iannis Xenakis (Ιωάννης Ιάννης Ξενάκης)]]></content:encoded>
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<title><![CDATA[Skyfire]]></title>
<link>http://bloggerschizo.net/2009/11/04/skyfire/</link>
<pubDate>Wed, 04 Nov 2009 19:23:45 +0000</pubDate>
<dc:creator>Özgür Özçınar</dc:creator>
<guid>http://bloggerschizo.net/2009/11/04/skyfire/</guid>
<description><![CDATA[Herkesçe bilinmek güzel olabilir! Selamlar Martin, umarım &#8220;Esoteric&#8221; albümü çıktıktan so]]></description>
<content:encoded><![CDATA[Herkesçe bilinmek güzel olabilir! Selamlar Martin, umarım &#8220;Esoteric&#8221; albümü çıktıktan so]]></content:encoded>
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<title><![CDATA[Spooky still!]]></title>
<link>http://padairvanvleck.wordpress.com/2009/11/02/spooky-still/</link>
<pubDate>Mon, 02 Nov 2009 09:00:18 +0000</pubDate>
<dc:creator>padairvanvleck</dc:creator>
<guid>http://padairvanvleck.wordpress.com/2009/11/02/spooky-still/</guid>
<description><![CDATA[HAIKU Dark silent phantoms        in the moon’s soft spectral light               grotesque tree sha]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>HAIKU</p>
<p>Dark silent phantoms</p>
<p>       in the moon’s soft spectral light</p>
<p>              grotesque tree shadows</p>
<p>3/16/90 Phyllis VanVleck</p>
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<title><![CDATA[RUBICON ASSAULT METAL NIGHT]]></title>
<link>http://hatchet88.wordpress.com/2009/10/27/rubicon-assault-metal-night/</link>
<pubDate>Tue, 27 Oct 2009 19:39:33 +0000</pubDate>
<dc:creator>hatchet88</dc:creator>
<guid>http://hatchet88.wordpress.com/2009/10/27/rubicon-assault-metal-night/</guid>
<description><![CDATA[Deci.. cu ce sa incep&#8230; CONCERT BAH CONCERT&#8230; cand? in curand!? Ce: RUBICON ASSAULT METAL ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Deci.. cu ce sa incep&#8230; CONCERT BAH CONCERT&#8230; cand? in curand!?</p>
<p>Ce: RUBICON ASSAULT METAL NIGHT!</p>
<p>Cine: 1. FACEPLANT (Metalcore &#8211; Timisoara)</p>
<p>2.<a href="http://www.myspace.com/spectralmetal"> SPECTRAL</a> (Technical Death Metal &#8211; Pitesti)</p>
<p>3. <a href="http://www.myspace.com/kultika">KULTIKA</a> (Thrash/Black/Sludge Metal &#8211; Timisoara)</p>
<p>4. <a href="http://www.myspace.com/bucovina">BUCOVINA</a> (Traditional/Folk Metal &#8211; Iasi)</p>
<p>Unde: Casa Studentilor Timisoara (B-dul Regele Carol, nr. 9);</p>
<p>Cand: 07 Noiembrie 2009 / 19:00;</p>
<p><strong><span style="text-decoration:line-through;">Cu aceasta ocazie, trupa Kultika isi va lansa primul EP auto-intitulat.</span></strong></p>
<p><strong>INTRAREA LIBERA!!!!!!!!</strong></p>
<p>Afisul:</p>
<p style="text-align:center;"><a rel="attachment wp-att-370" href="http://hatchet88.wordpress.com/2009/10/27/rubicon-assault-metal-night/afis/"><img class="aligncenter size-full wp-image-370" title="afis" src="http://hatchet88.wordpress.com/files/2009/10/afis.jpg" alt="afis" width="540" height="763" /></a></p>
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<title><![CDATA[Potopul din capitala]]></title>
<link>http://hatchet88.wordpress.com/2009/10/15/potopul-din-capitala/</link>
<pubDate>Thu, 15 Oct 2009 16:26:43 +0000</pubDate>
<dc:creator>hatchet88</dc:creator>
<guid>http://hatchet88.wordpress.com/2009/10/15/potopul-din-capitala/</guid>
<description><![CDATA[Scriu din Bucuresti! Ar trebui sa fie un motiv de lauda? Chiar nu, dar na, scriu din Bucuresti]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Scriu din Bucuresti! Ar trebui sa fie un motiv de lauda? Chiar nu, dar na, scriu din Bucuresti&#8230; simt ceva diferit fata de scrisul din Timisoara??? Poate mai multa greata, dar asta probabil de la tigara cu care imi inec plamanii. Azi dimineata am pornit din Cluj-Napoca, in goana, la volan&#8230;si&#8230; am evitat cateva accidente pe Valea Oltului, ca doar na, tineam la pedala cativa cai frumosi si artagosi. Drumul a fost linistit/linistitor (mai mult cred ca as tinde spre cuvantul plictisitor). Am vazut prima zapada de anu asta, am atins prima zapada de anu asta (sac &#8211; sac). Dupa o scurta oprire in Sibiu, doar nu o sa circulam cu stomacocul golut <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_razz.gif' alt=':P' class='wp-smiley' /> , am pornit in goana, cu aceeasi cai frumosi si neobositi. Dupa 45 de minute pe autostrada am ajuns la captul Bucurestiul, capatul ala care intra in Bucuresti si dupa 80 de minute, am cronometrat frateeeeeloooo, am ajuns in centru. Fain. E a 3-a oara cand fac drumul acesta in ultimele 4 saptamani, deja e prea mult. M-am saturat, m-am plictisit si de Bucurestiul Vechi. Parca dupa ce ating de 2-3 ori locurile prin care umblu, se duce dracu toata frumusetea. Maine cred ca dau o fuga pana Muzica sa-mi cumpar niste ate la bas. Simt eu ca am nevoie de o schimbare in sunet. Cam atat din centrul Bucurestiului.<br />
Acum partea interesanta: &#8220;se zvoneste prin ziare&#8221;,mai mult plin glasul soptit de la ureche, ca o sa avem un  concert care se apropie cu pasi repejori <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_biggrin.gif' alt=':D' class='wp-smiley' /> . How Cul Is That?!<br />
-Unde : Casa Studentilor<br />
-Cand : Revin, dar stiu ceva de ora 20:00<br />
-Cine : FacePlant, Spectral, Kultika, Bucovina<br />
-Cat : Cica gratis frate&#8230; dar revin cu detalii<br />
-Cum : hmmm??? Trick question. Making noizzzzz?<br />
-De ce : Asta nu stiu, dar poate ca era cazul sa se faca o chestie din asta?<br />
-Ce : Muzica, din astea, cum putini cunosc (mai nou)<br />
-Altceva : Ma doare pateul!<br />
Numa bine bai, si sa ne auzim cu bine si toate cele&#8230;<br />
<span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/GHPixYiiNG8&#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/GHPixYiiNG8&#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>
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<title><![CDATA[Four CCP Records titles available for pre-orders]]></title>
<link>http://lethalconflict.wordpress.com/2009/10/14/four-ccp-records-titles-available-for-pre-orders/</link>
<pubDate>Wed, 14 Oct 2009 14:24:47 +0000</pubDate>
<dc:creator>lethalconflict</dc:creator>
<guid>http://lethalconflict.wordpress.com/2009/10/14/four-ccp-records-titles-available-for-pre-orders/</guid>
<description><![CDATA[http://www.lethalconflict.com 13.10.2009 PRE-ORDER until 30.11.2009 Atritas &#8211; Celestial Decay ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><a title="Lethal Conflict Mailorder" href="http://www.lethalconflict.com" target="_blank">http://www.lethalconflict.com</a> 13.10.2009</p>
<p>PRE-ORDER until <strong>30.11.2009</strong></p>
<p><strong>Atritas</strong> &#8211; Celestial Decay CD € 9.80 / Ft 2490<br />
Swizz melodic black metal<br />
<a title="http://www.atritas.ch" href="http://www.atritas.ch" target="_blank">http://www.atritas.ch</a><br />
<a title="http://www.last.fm/music/Atritas " href="http://www.last.fm/music/Atritas" target="_blank">http://www.last.fm/music/Atritas </a><br />
<strong>Auringon Hauta</strong> &#8211; Muinaisia Muisteloita, Noita Syntyjä Syviä CD € 9.80 / Ft 2490<br />
Finnish folk/heathen metal, ex-Solgrav<br />
<a title="http://www.metal-archives.com/band.php?id=11135" href="http://www.metal-archives.com/band.php?id=11135" target="_blank">http://www.metal-archives.com/band.php?id=11135</a><br />
<a title="http://www.last.fm/music/Solgrav " href="http://www.last.fm/music/Solgrav" target="_blank">http://www.last.fm/music/Solgrav </a><br />
<strong>Spectral</strong> &#8211; Evil Iron Kingdom CD € 9.80 / Ft 2490<br />
German viking/battle/power metal<br />
<a title="http://www.myspace.com/blackvikingpower " href="http://www.myspace.com/blackvikingpower" target="_blank">http://www.myspace.com/blackvikingpower </a><br />
<a title="http://www.last.fm/music/Spectral " href="http://www.last.fm/music/Spectral" target="_blank">http://www.last.fm/music/Spectral </a><br />
<strong>Woodland </strong>- Dreamality CD € 9.80 / Ft 2490<br />
German folk/dark metal<br />
<a title="http://www.myspace.com/metalwoodland" href="http://www.myspace.com/metalwoodland" target="_blank">http://www.myspace.com/metalwoodland</a><br />
<a title="http://www.last.fm/music/Woodland" href="http://www.last.fm/music/Woodland" target="_blank">http://www.last.fm/music/Woodland</a></p>
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<title><![CDATA[Ghosts of Techno]]></title>
<link>http://astrangelyisolatedplace.wordpress.com/2009/10/12/ghosts-of-techno/</link>
<pubDate>Mon, 12 Oct 2009 09:16:54 +0000</pubDate>
<dc:creator>astrangelyisolatedplace</dc:creator>
<guid>http://astrangelyisolatedplace.wordpress.com/2009/10/12/ghosts-of-techno/</guid>
<description><![CDATA[A nice little insight into one of my favourite labels, Ghostly International, home of Tycho and Lusi]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>A nice little insight into one of my favourite labels, <a href="http://ghostly.com/" target="_blank">Ghostly International</a>, home of <a href="http://blog.iso50.com/" target="_blank">Tycho</a> and Lusine. (<a href="http://surfingonsteam.blogspot.com/2009/10/current-shines-light-on-ghostly.html" target="_blank">via Surfing On Steam</a>)</p>
<p><code><span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/e3R8eZbGZ80&#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/e3R8eZbGZ80&#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></code></p>
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<title><![CDATA[spectral signatures of life ONE]]></title>
<link>http://dslrastrophotography.wordpress.com/2009/10/06/spectral-signatures-of-life-one/</link>
<pubDate>Tue, 06 Oct 2009 20:00:04 +0000</pubDate>
<dc:creator>woogunner</dc:creator>
<guid>http://dslrastrophotography.wordpress.com/2009/10/06/spectral-signatures-of-life-one/</guid>
<description><![CDATA[Dr. Lisa Kaltenegger Friday, June 1, 2007, 7:30 pm at Seagrave Observatory www.theskyscrapers.org In]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Dr. Lisa Kaltenegger Friday, June 1, 2007, 7:30 pm at Seagrave Observatory www.theskyscrapers.org In a famous paper, Sagan analyzed a spectrum of the Earth taken by the Galileo probe, searching for signatures of life. They concluded that the large amount of oxygen and the simultaneous presence of methane traces are suggestive of biology. In this talk we discuss biomarkers and focus on what makes a habitable planet, using Earth as our example. What can we see? How can we detect those faint &#8230;</p>
<p><span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/YWYggZEefgI&#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/YWYggZEefgI&#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>
<p><a href='http://www.youtube.com/watch?v=YWYggZEefgI&#38;hl=en' rel='nofollow'>http://www.youtube.com/watch?v=YWYggZEefgI&#38;hl=en</a>
<p>Friends Link : Best HDTV  <a href="http://www.besthdtv2you.com" rel="dofollow" title="">Best HDTV For You</a>  <a href="http://vgahdtvconverter.blogspot.com" rel="dofollow" title="VGA HDTV Converters">VGA HDTV Converters</a>  <a href="http://www.cleverclickonline.com" rel="dofollow" title="Best Budget DSLR">Best Budget DSLR</a>  <a href="http://followfridaytwitter.blogspot.com" rel="dofollow" title="Follow Friday Twitter">Follow Friday Twitter</a> </p>
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<title><![CDATA[New Release]]></title>
<link>http://ellisonotes.wordpress.com/2009/10/04/new-release/</link>
<pubDate>Sun, 04 Oct 2009 13:11:13 +0000</pubDate>
<dc:creator>ellisonotes</dc:creator>
<guid>http://ellisonotes.wordpress.com/2009/10/04/new-release/</guid>
<description><![CDATA[In early 2010, Innova Recordings will release Invocation, which includes the Borromeo Quartet&#8217;]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>In early 2010, Innova Recordings will release <em>Invocation</em>, which includes the Borromeo Quartet&#8217;s masterful rendition of Ellison&#8217;s seven-movement String Quartet #2, Helen Bledsoe&#8217;s solo &#8216;extended&#8217; flute on <em>Invocation-Meditation-Allegro</em>, and the historical live recording at the 2003 Istanbul International Spectral Music Conference of the blind master Turkish Hafız, Kani Karaca, peforming Ellison&#8217;s exploration into new worlds of vocal sound and instrumental combination, <em>Elif.</em>  This concert was the second-to-last concert and very last recording of Karaca, widely considered one of the two most influential master Turkish singers of the 20th century.  Distribution by Naxos.</p>
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<title><![CDATA[A simple remark on the spectrum of certain matrix sums]]></title>
<link>http://ifwisdomwereteachable.wordpress.com/2009/10/02/a-simple-remark-on-the-spectrum-of-certain-matrix-sums/</link>
<pubDate>Fri, 02 Oct 2009 05:39:28 +0000</pubDate>
<dc:creator>toomuchcoffeeman</dc:creator>
<guid>http://ifwisdomwereteachable.wordpress.com/2009/10/02/a-simple-remark-on-the-spectrum-of-certain-matrix-sums/</guid>
<description><![CDATA[Having let the Shavgulidze-Thompson project slide out of the intray and into the mountain of Unfinis]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Having let the Shavgulidze-Thompson project slide out of the intray and into the mountain of Unfinished Loose Ends, I feel I should compensate with something vaguely mathematical. Hence this post, which is a follow up to some comments I left on a <a href="http://sbseminar.wordpress.com/2009/09/29/the-20-questions-seminar/">post at the Secret Blogging Seminar.</a></p>
<p>More precisely, in response to Q2 on that post, I left some rather <a href="http://sbseminar.wordpress.com/2009/09/29/the-20-questions-seminar/#comment-6642">dim-witted</a> and <a href="http://sbseminar.wordpress.com/2009/09/29/the-20-questions-seminar/#comment-6644">error-strewn comments</a>, only to have light shed by this subsequent <a href="http://sbseminar.wordpress.com/2009/09/29/the-20-questions-seminar/#comment-6646">observation from Greg Kuperberg:</a></p>
<p><b>Proposition:</b> <i>Let A and B be two Hermitian matrices. Then the spectrum of A+iB lies in the rectangle formed by the first and last eigenvalues of A and B.</i></p>
<p>Once GK stated the correct result, I realised that it followed from some facts that I really should have known &#8211; or knew, but had momentarily forgotten. It seems that the argument I had in mind is slightly different, at least in presentation, from the <a href="http://sbseminar.wordpress.com/2009/09/29/the-20-questions-seminar/#comment-6694">proof GK had in mind</a>, and so I thought I&#8217;d give it here. (His reasoning seems like it should be more robust, and extend more easily to the case of bounded operators on infinite-dimensional Hilbert space.)</p>
<p><b>Claim.</b> <i>Let <img src='http://l.wordpress.com/latex.php?latex=A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='B' title='B' class='latex' /> be normal matrices, with spectra <img src='http://l.wordpress.com/latex.php?latex=%5Csigma%28A%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\sigma(A)' title='\sigma(A)' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=%5Csigma%28B%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\sigma(B)' title='\sigma(B)' class='latex' /> respectively. Then the spectrum of <img src='http://l.wordpress.com/latex.php?latex=A%2BB&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A+B' title='A+B' class='latex' /> is contained in <img src='http://l.wordpress.com/latex.php?latex=%7B%5Crm+co%7D%5C%2C+%5Csigma%28A%29%2B%7B%5Crm+co%7D%5C%2C%5Csigma%28B%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\rm co}\, \sigma(A)+{\rm co}\,\sigma(B)' title='{\rm co}\, \sigma(A)+{\rm co}\,\sigma(B)' class='latex' />.</i></p>
<p><b>Proof.</b> Since <img src='http://l.wordpress.com/latex.php?latex=A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' /> is normal, there exists an orthonormal basis of <img src='http://l.wordpress.com/latex.php?latex=%7B%5Cbf+C%7D%5En&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='{\bf C}^n' title='{\bf C}^n' class='latex' />, which consists of eigenvectors for <img src='http://l.wordpress.com/latex.php?latex=A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' />. Let&#8217;s denote this basis by <img src='http://l.wordpress.com/latex.php?latex=v_1%2C%5Cdots%2Cv_n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='v_1,\dots,v_n' title='v_1,\dots,v_n' class='latex' /> and let the corresponding eigenvalues be <img src='http://l.wordpress.com/latex.php?latex=%5Clambda_1%2C%5Cdots%2C%5Clambda_n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\lambda_1,\dots,\lambda_n' title='\lambda_1,\dots,\lambda_n' class='latex' />.</p>
<p>Similarly, there is an orthonormal basis <img src='http://l.wordpress.com/latex.php?latex=w_1%2C%5Cdots%2C+w_n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w_1,\dots, w_n' title='w_1,\dots, w_n' class='latex' /> and scalars <img src='http://l.wordpress.com/latex.php?latex=%5Cmu_1%2C%5Cdots%2C%5Cmu_n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\mu_1,\dots,\mu_n' title='\mu_1,\dots,\mu_n' class='latex' /> such that <img src='http://l.wordpress.com/latex.php?latex=Bw_k%3D%5Cmu_kw_k&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='Bw_k=\mu_kw_k' title='Bw_k=\mu_kw_k' class='latex' /> for all <img src='http://l.wordpress.com/latex.php?latex=j&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='j' title='j' class='latex' />.</p>
<p>Now let <img src='http://l.wordpress.com/latex.php?latex=%5Calpha&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\alpha' title='\alpha' class='latex' /> be an eigenvalue of <img src='http://l.wordpress.com/latex.php?latex=A%2BB&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A+B' title='A+B' class='latex' />, and let <img src='http://l.wordpress.com/latex.php?latex=x&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x' title='x' class='latex' /> be a corresponding eigenvector of unit length. We have</p>
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Calpha+%3D+%5Clangle+%5Calpha+x+%2C+x%5Crangle+%3D+%5Clangle+%28A%2BB%29x%2Cx%5Crangle+%3D+%5Clangle+Ax%2Cx%5Crangle+%2B+%5Clangle+Bx%2C+x%5Crangle+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\alpha = \langle \alpha x , x\rangle = \langle (A+B)x,x\rangle = \langle Ax,x\rangle + \langle Bx, x\rangle ' title='\alpha = \langle \alpha x , x\rangle = \langle (A+B)x,x\rangle = \langle Ax,x\rangle + \langle Bx, x\rangle ' class='latex' /></p>
<p>But now we can exploit the fact that <img src='http://l.wordpress.com/latex.php?latex=A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='B' title='B' class='latex' /> each have a complete set of orthonormal eigenvectors. In particular, writing <img src='http://l.wordpress.com/latex.php?latex=x+%3D+%5Csum_j+%5Clangle+x%2C+v_j+%5Crangle+v_j&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x = \sum_j \langle x, v_j \rangle v_j' title='x = \sum_j \langle x, v_j \rangle v_j' class='latex' /> we have</p>
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Clangle+Ax+%2C+x+%5Crangle+%3D+%5Csum_j+%5Clambda_j+%5Cvert+%5Clangle+x%2C+v_j+%5Crangle+%5Cvert%5E2+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\langle Ax , x \rangle = \sum_j \lambda_j \vert \langle x, v_j \rangle \vert^2 ' title='\langle Ax , x \rangle = \sum_j \lambda_j \vert \langle x, v_j \rangle \vert^2 ' class='latex' /></p>
<p>We have <img src='http://l.wordpress.com/latex.php?latex=%5Csum_j+%5Cvert%5Clangle+x%2C+v_j%5Crangle%5Cvert%5E2%3D1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\sum_j \vert\langle x, v_j\rangle\vert^2=1' title='\sum_j \vert\langle x, v_j\rangle\vert^2=1' class='latex' /> (again, using the orthonormality of the <img src='http://l.wordpress.com/latex.php?latex=v_j&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='v_j' title='v_j' class='latex' />) and so <img src='http://l.wordpress.com/latex.php?latex=%5Clangle+Ax%2Cx%5Crangle+%5Cin+%7B%5Crm+co%7D%5C%2C+%5C%7B+%5Clambda_1%2C%5Cdots%2C%5Clambda_n%5C%7D+%3D+%7B%5Crm+co%7D%5Csigma%28A%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\langle Ax,x\rangle \in {\rm co}\, \{ \lambda_1,\dots,\lambda_n\} = {\rm co}\sigma(A)' title='\langle Ax,x\rangle \in {\rm co}\, \{ \lambda_1,\dots,\lambda_n\} = {\rm co}\sigma(A)' class='latex' />. An exactly similar argument, this time using the <img src='http://l.wordpress.com/latex.php?latex=w_k&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='w_k' title='w_k' class='latex' />, tells us that <img src='http://l.wordpress.com/latex.php?latex=%5Clangle+Bx%2C+x%5Crangle+%5Cin+%7B%5Crm+co%7D%5C%2C%5Csigma%28B%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\langle Bx, x\rangle \in {\rm co}\,\sigma(B)' title='\langle Bx, x\rangle \in {\rm co}\,\sigma(B)' class='latex' />. Hence <img src='http://l.wordpress.com/latex.php?latex=%5Calpha%3D%5Clangle+Ax%2Cx%5Crangle%2B+%5Clangle+Bx%2Cx%5Crangle&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\alpha=\langle Ax,x\rangle+ \langle Bx,x\rangle' title='\alpha=\langle Ax,x\rangle+ \langle Bx,x\rangle' class='latex' /> lies in the sum of these two convex hulls, as claimed.</p>
<h2>Cards on the table, or the man behind the curtain</h2>
<p>I have to confess that the phrasing of the argument above wasn&#8217;t the first that came to mind when I read GK&#8217;s comment. Lurking in the background &#8212; above and, I suspect, in his approach also &#8212;  is the concept of <i>numerical range.</i> The numerical range of an <img src='http://l.wordpress.com/latex.php?latex=n%5Ctimes+n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n\times n' title='n\times n' class='latex' /> complex matrix <img src='http://l.wordpress.com/latex.php?latex=M&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='M' title='M' class='latex' /> is the set</p>
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=W%28A%29%3D%5C%7B+%5Clangle+Mx+%2Cx+%5Crangle+%26%23124%3B+x+%5Cin+%7B%5Cbf+C%7D%5En%2C+%5C%26%23124%3Bx%5C%26%23124%3B_2+%5Cleq+1+%5C%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(A)=\{ \langle Mx ,x \rangle &#124; x \in {\bf C}^n, \&#124;x\&#124;_2 \leq 1 \} ' title='W(A)=\{ \langle Mx ,x \rangle &#124; x \in {\bf C}^n, \&#124;x\&#124;_2 \leq 1 \} ' class='latex' /></p>
<p>and it is clear that <img src='http://l.wordpress.com/latex.php?latex=W%28A%2BB%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(A+B)' title='W(A+B)' class='latex' /> is contained in <img src='http://l.wordpress.com/latex.php?latex=W%28A%29%2BW%28B%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(A)+W(B)' title='W(A)+W(B)' class='latex' /> for every pair <img src='http://l.wordpress.com/latex.php?latex=A%2C+B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A, B' title='A, B' class='latex' /> of <img src='http://l.wordpress.com/latex.php?latex=n%5Ctimes+n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n\times n' title='n\times n' class='latex' /> matrices. Now, by considering appropriate eigenvectors, one sees that every eigenvalue of <img src='http://l.wordpress.com/latex.php?latex=M&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='M' title='M' class='latex' /> is contained in <img src='http://l.wordpress.com/latex.php?latex=W%28M%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(M)' title='W(M)' class='latex' />. Also, if <img src='http://l.wordpress.com/latex.php?latex=D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='D' title='D' class='latex' /> is a diagonal matrix, then the same calculation that was made above shows that <img src='http://l.wordpress.com/latex.php?latex=W%28D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(D)' title='W(D)' class='latex' /> is contained in the convex hull of <img src='http://l.wordpress.com/latex.php?latex=%5Csigma%28D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\sigma(D)' title='\sigma(D)' class='latex' />, and since the numerical range is unchanged if we conjugate by a unitary matrix, it follows that <img src='http://l.wordpress.com/latex.php?latex=W%28M%29%5Csubseteq+%7B%5Crm+co%7D%5C%2C+%5Csigma%28M%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(M)\subseteq {\rm co}\, \sigma(M)' title='W(M)\subseteq {\rm co}\, \sigma(M)' class='latex' /> for every <i>normal</i> matrix <img src='http://l.wordpress.com/latex.php?latex=M&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='M' title='M' class='latex' />. In particular, if <img src='http://l.wordpress.com/latex.php?latex=A&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='A' title='A' class='latex' /> and <img src='http://l.wordpress.com/latex.php?latex=B&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='B' title='B' class='latex' /> are normal <img src='http://l.wordpress.com/latex.php?latex=n%5Ctimes+n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n\times n' title='n\times n' class='latex' /> matrices then</p>
<p align="center"><img src='http://l.wordpress.com/latex.php?latex=%5Csigma%28A%2BB%29+%5Csubseteq+W%28A%2BB%29+%5Csubseteq+W%28A%29%2BW%28B%29+%3D+%7B%5Crm+co%7D%5C%2C%5Csigma%28A%29%2B%7B%5Crm+co%7D%5C%2C%5Csigma%28B%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\sigma(A+B) \subseteq W(A+B) \subseteq W(A)+W(B) = {\rm co}\,\sigma(A)+{\rm co}\,\sigma(B)' title='\sigma(A+B) \subseteq W(A+B) \subseteq W(A)+W(B) = {\rm co}\,\sigma(A)+{\rm co}\,\sigma(B)' class='latex' /></p>
<p>which is in effect what we proved above.</p>
<p>The reason I should have remembered this is that a short while back I was reading up on some aspects of the numerical range for operators on infinite-dimensional spaces. The definition is the obvious one, and what is interesting is that we still have</p>
<ol>
<li> <img src='http://l.wordpress.com/latex.php?latex=%5Csigma%28T%29%5Csubseteq+W%28T%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\sigma(T)\subseteq W(T)' title='\sigma(T)\subseteq W(T)' class='latex' /> for every bounded operator <img src='http://l.wordpress.com/latex.php?latex=T&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='T' title='T' class='latex' />;</li>
<li> <img src='http://l.wordpress.com/latex.php?latex=W%28M%29+%3D+%7B%5Crm+co%7D%5C%2C%5Csigma%28M%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='W(M) = {\rm co}\,\sigma(M)' title='W(M) = {\rm co}\,\sigma(M)' class='latex' /> for every normal operator <img src='http://l.wordpress.com/latex.php?latex=M&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='M' title='M' class='latex' />.</li>
</ol>
<p>Note that in infinite dimensions the spectrum of <img src='http://l.wordpress.com/latex.php?latex=T&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='T' title='T' class='latex' /> might contain points which are not eigenvalues, and so the argument above with eigenvectors doesn&#8217;t work anymore.</p>
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<title><![CDATA[Sound Analysis - Assignment 3]]></title>
<link>http://soundsofstanford.wordpress.com/2009/09/11/sound-analysis-assignment-3/</link>
<pubDate>Fri, 11 Sep 2009 16:04:33 +0000</pubDate>
<dc:creator>roofdog123</dc:creator>
<guid>http://soundsofstanford.wordpress.com/2009/09/11/sound-analysis-assignment-3/</guid>
<description><![CDATA[(1) Ocarina (Harmonic and stable) In looking for a sustained and harmonic note, the Ocarina&#8217;s ]]></description>
<content:encoded><![CDATA[(1) Ocarina (Harmonic and stable) In looking for a sustained and harmonic note, the Ocarina&#8217;s ]]></content:encoded>
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<title><![CDATA[Aussie Scientists Pack Massive Data Onto Single 5-D Discs]]></title>
<link>http://saliaziz.wordpress.com/2009/08/29/aussie-scientists-pack-massive-data-onto-single-5-d-discs/</link>
<pubDate>Fri, 28 Aug 2009 19:04:54 +0000</pubDate>
<dc:creator>Syed Ali Aziz</dc:creator>
<guid>http://saliaziz.wordpress.com/2009/08/29/aussie-scientists-pack-massive-data-onto-single-5-d-discs/</guid>
<description><![CDATA[Aussies develop multi-terabyte optical storage&#8230; Researchers at a University in Melbourne, Aust]]></description>
<content:encoded><![CDATA[Aussies develop multi-terabyte optical storage&#8230; Researchers at a University in Melbourne, Aust]]></content:encoded>
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<title><![CDATA[Gerard Grisey's Prologue]]></title>
<link>http://monashcomposers.wordpress.com/2009/08/15/gerard-griseys-prologue/</link>
<pubDate>Sat, 15 Aug 2009 00:54:30 +0000</pubDate>
<dc:creator>tjlaw7</dc:creator>
<guid>http://monashcomposers.wordpress.com/2009/08/15/gerard-griseys-prologue/</guid>
<description><![CDATA[I suspect that if I had listened to this piece about 4 months ago, I would have disliked it &#8211; ]]></description>
<content:encoded><![CDATA[I suspect that if I had listened to this piece about 4 months ago, I would have disliked it &#8211; ]]></content:encoded>
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<title><![CDATA[I Am The Car]]></title>
<link>http://marcelo717.wordpress.com/2009/07/25/i-am-the-car/</link>
<pubDate>Sat, 25 Jul 2009 13:21:36 +0000</pubDate>
<dc:creator>Castro</dc:creator>
<guid>http://marcelo717.wordpress.com/2009/07/25/i-am-the-car/</guid>
<description><![CDATA[I went for a drive today&#8230;   Track is I Am The Car by Audion]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>I went for a drive today&#8230;</p>
<p><span style='text-align:center; display: block;'><object width='425' height='350'><param name='movie' value='http://www.youtube.com/v/Q1U4ln5I0mI&#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/Q1U4ln5I0mI&#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>
<p> </p>
<p>Track is I Am The Car by Audion</p>
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<title><![CDATA[The Sound of Light]]></title>
<link>http://blog.imaging-git.com/2009/07/01/the-sound-of-light/</link>
<pubDate>Wed, 01 Jul 2009 14:29:42 +0000</pubDate>
<dc:creator>aszerdi</dc:creator>
<guid>http://blog.imaging-git.com/2009/07/01/the-sound-of-light/</guid>
<description><![CDATA[Together with his research team, Professor Vasilis Ntziachristos from the Helmholtz Zentrum Munich, ]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p style="text-align:justify;">Together with his research team, Professor Vasilis Ntziachristos from the Helmholtz Zentrum Munich, Germany and the Technical University Munich, Germany developed a new technology to make light audible. The technique, called multi-spectral opto-acoustic tomography (MSOT), combines light and ultrasound to visualize fluorescent proteins that are seated several centimeters deep into living tissue.<br />
The researchers used a genetically modified adult zebra fish which carried fluorescent pigments in its tissue. They illuminated the fish from multiple angles using flashes of laser light that are absorbed by the fluorescent pigments in the fish. The pigments absorb the light, a process that causes slight local increases of temperature, which in turn result in tiny local volume expansions. This happens very quickly and creates small shock waves. In effect, the short laser pulse gives rise to an ultrasound wave that the researchers pick up with an ultrasound microphone. To analyze the resulting acoustic patterns, a computer is attached. The computer uses specially developed mathematical formulas to evaluate and interpret the specific distortions caused by scales, muscles, bones and internal organs to generate a three-dimensional image. In the future this technology may facilitate the examination of tumors or coronary vessels in humans.<br />
<a href="http://www.helmholtz-muenchen.de/en/press-and-media/press-releases/press-releases-2009/press-releases-2009-detail/article/12021/44/index.html" target="_blank">www.helmholtz-muenchen.de/en</a></p>
<p style="text-align:justify;">
<p style="text-align:justify;">
<div id="attachment_981" class="wp-caption aligncenter" style="width: 460px"><img class="size-full wp-image-981" title="zebra fish" src="http://imaginggit.wordpress.com/files/2009/07/zebra-fish.jpg" alt="Multi-spectral opto-acoustic tomography or MSOT allows the investigation of subcellular processes in live organisms." width="450" height="160" /><p class="wp-caption-text">Multi-spectral opto-acoustic tomography or MSOT allows the investigation of subcellular processes in live organisms.</p></div>
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<title><![CDATA[Kate Simko - Take you there Ep (Spectral Sound Us)]]></title>
<link>http://badenprojekt.wordpress.com/2009/06/25/kate-simko-take-you-there-ep-spectral-sound-us/</link>
<pubDate>Thu, 25 Jun 2009 11:28:48 +0000</pubDate>
<dc:creator>badenprojekt</dc:creator>
<guid>http://badenprojekt.wordpress.com/2009/06/25/kate-simko-take-you-there-ep-spectral-sound-us/</guid>
<description><![CDATA[Perfect for a tech-housy atmosphere&#8230; Remix by Bruno Pronsato is sick Tracklist: A1 &#8220;Take]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><a href="http://www.katesimko.com/"><img class="alignnone size-full wp-image-121" title="Kate simko" src="http://badenprojekt.wordpress.com/files/2009/06/cs358356-01a-big.jpg" alt="Kate simko" width="450" height="456" /></a></p>
<p>Perfect for a tech-housy atmosphere&#8230; Remix by <strong>Bruno Pronsato</strong> is sick</p>
<p><strong>Tracklist:</strong></p>
<p>A1 &#8220;Take You There&#8221; (feat Brenda D)<br />
A2 &#8220;Down Beat&#8221;<br />
B1 &#8220;Take You There&#8221; (feat Brenda D &#8211; Bruno Pronsato Break-Up Day)<br />
B2 &#8220;Margie&#8217;&#8217;s Groove&#8221;</p>
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<title><![CDATA[You Should Not be Ghost Hunting if....]]></title>
<link>http://meadmuse.wordpress.com/2009/06/08/you-should-not-be-ghost-hunting-if/</link>
<pubDate>Mon, 08 Jun 2009 04:52:47 +0000</pubDate>
<dc:creator>meadmuse</dc:creator>
<guid>http://meadmuse.wordpress.com/2009/06/08/you-should-not-be-ghost-hunting-if/</guid>
<description><![CDATA[Working with the spirits ~ Wooooohhhhhh! Let&#8217;s start off with the discussion of who should NOT]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p>Working with the spirits ~ Wooooohhhhhh!</p>
<p>Let&#8217;s start off with the discussion of who should NOT be messing with the spirits.</p>
<p>1. Anyone who does not understand and grasp the concept that the term &#8220;spirits&#8221; applies to a plethora of entities and does not have a basic understanding of what they might be. Ignorance may be bliss in the mortal world, but it can be dangerous in spirit workings.</p>
<p>2. Anyone who gets the willies easy and has an extreme physical reaction such as fainting.</p>
<p>Puking is all right. You will stop puking eventually. But not much you can do if you pass out, eh? But remember to be polite and not breathe on anyone standing near you. Smells bad.</p>
<p>Oh, yeah, runners kind of suck. Not so much that they take the fight or flight thing seriously, but they seldom fight and only run over the top of you and knock you down. Not fun plus you get scrapped knees.</p>
<p>Oh, yeah, grabbers too! Almost forgot about the grabbers who crawl on top of your back and wrap their arms around your neck so you can&#8217;t breathe. Not so much fun either.</p>
<p>You can hide behind someone. That&#8217;s okay. But don&#8217;t block the exit in case they are a runner or they will go over and through you and leave you there with scrapped knees.</p>
<p>And a little side note, any unusual occurrences in your panties are forgivable PROVIDED YOU DON&#8217;T BRING ATTENTION TO IT! Really! I mean it! I have a right to plausible deniability that there is something icky is in someone else&#8217;s pants and I am fully prepared to pretend it didn&#8217;t happen. </p>
<p>3. Anyone who wants to poke the dead dudes on national television, get them all riled up and then leave the poor property owners with pissed off spirits. Come on, guys. not nice and some of us are getting tired of cleaning up the mess you left behind.</p>
<p>4. Anyone who thinks that every spirit is a demon or the &#8220;Great Dark One of the Great Foreverness Pit of Despair and Doom&#8221; or whatever other title that has chosen &#8220;them&#8221; out of millions of other far more interesting people to annoy. Not sure what to say other than, boy you must really believe you are all that and a bag of chips to glean that much attention.</p>
<p>5. Anyone who slept through or drew doodle picture during basic self-protection 101 or still needs notes to remember how to do it.</p>
<p>6. Anyone who thinks it never stops being funny jumping out of closets and saying boogity boogity boo for the 9,364th time in the past hour. </p>
<p><em>Reposting of a discussion I posted on Paganspace.  10/25/08</em></p>
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<title><![CDATA[Audion - Raw Dog
]]></title>
<link>http://feelmybicep.com/2009/05/23/audion-raw-dog/</link>
<pubDate>Sat, 23 May 2009 13:44:06 +0000</pubDate>
<dc:creator>feelmybicep</dc:creator>
<guid>http://feelmybicep.com/2009/05/23/audion-raw-dog/</guid>
<description><![CDATA[Lets keep this techno shit rolling. I was loving this record when I got it, such respectable electri]]></description>
<content:encoded><![CDATA[<div class='snap_preview'><p><img class="aligncenter" src="http://ugliest-dog.org/images/60.jpg" alt="" width="400" height="324" /></p>
<p>Lets keep this techno shit rolling. I was loving this record when I got it, such respectable electric dirtiness. Matthew Dear is really rather great, maybe not everyone&#8217;s favourite but his production is mondo tight. Even Asa Breed supplies whats needed, dance music for all sorts of folk.</p>
<p><span style='text-align:left;display:block;'><p><object type='application/x-shockwave-flash' data='http://wordpress.com/wp-content/plugins/audio-player/player.swf' width='290' height='24' id='audioplayer1'><param name='movie' value='http://wordpress.com/wp-content/plugins/audio-player/player.swf' /><param name='FlashVars' value='&amp;bg=0xf8f8f8&amp;leftbg=0xeeeeee&amp;lefticon=0x666666&amp;rightbg=0xcccccc&amp;rightbghover=0x999999&amp;righticon=0x666666&amp;righticonhover=0xffffff&amp;text=0x666666&amp;slider=0x666666&amp;track=0xFFFFFF&amp;border=0x666666&amp;loader=0x9FFFB8&amp;soundFile=http%3A%2F%2Fdl.getdropbox.com%2Fu%2F254919%2FBICEP%2520May%2Fd1-audion--raw_dog-cmc.mp3' /><param name='quality' value='high' /><param name='menu' value='false' /><param name='bgcolor' value='#FFFFFF' /></object></p></span></p>
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<pubDate>Thu, 30 Apr 2009 10:42:55 +0000</pubDate>
<dc:creator>vhbelvadi</dc:creator>
<guid>http://thewriteeye.wordpress.com/2009/04/30/the-big-bang-theory-in-depth-part-i/</guid>
<description><![CDATA[In my last post, I gave &#8211; what I believe to be &#8211; a trivial point against the big bang th]]></description>
<content:encoded><![CDATA[In my last post, I gave &#8211; what I believe to be &#8211; a trivial point against the big bang th]]></content:encoded>
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<title><![CDATA[Alexi Delano &amp; Xpansul - 2nite @ Milan, Italy]]></title>
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<guid>http://fortresstakes.wordpress.com/2009/04/08/visionaries-knights-of-the-magical-light-1987-1-season-13-episodes/</guid>
<description><![CDATA[The animation boom of the 1980&#8217;s often will bring a wistful smile to any who experienced it.  ]]></description>
<content:encoded><![CDATA[The animation boom of the 1980&#8217;s often will bring a wistful smile to any who experienced it.  ]]></content:encoded>
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