tag:blogger.com,1999:blog-37595492024-03-08T00:18:49.699+05:30Arbit ScratchpadIn my mind is a strange attractor!shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.comBlogger551125tag:blogger.com,1999:blog-3759549.post-73516286236482293232022-04-02T04:05:00.009+05:302022-04-02T04:54:06.891+05:30CODA: Art & Ambiguity<p></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/VtfWXtHKBno" width="469" youtube-src-id="VtfWXtHKBno"></iframe></div><br />All good art has something ambiguous about it, that gives the audience or the viewer an impetus to interpret it herself! This particular scene from the fantastic movie CODA (Oscar winner 2022) was exactly like that! In it a deaf father's daughter is singing a song for him and he is able to feel it. The second time I watched the movie, I got a totally different interpretation than the way I had interpreted it, this time from Tanuka who was watching it for the first time. She had interpreted it in a way I just did not think of, without going into specifics. <div><br /></div><div>Which goes on to show the element of ambiguity in that scene which throws up a moment of aha! Or you got to use your brain to fill in the blanks and give it closure. Of course there is context and emotional depth that is necessary too, however the hallmark of good film-making are such subtly ambiguous moments and scenes without being over the top irrational or puzzling and uninterpretable! Good art, ergo the subtle ambiguity! <p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://media.npr.org/assets/img/2021/08/12/coda_photo_0106_wide-70d27a3a56ce1656824675ec7e5e017af621da4d-s800-c85.webp" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="449" data-original-width="800" height="349" src="https://media.npr.org/assets/img/2021/08/12/coda_photo_0106_wide-70d27a3a56ce1656824675ec7e5e017af621da4d-s800-c85.webp" width="622" /></a></div><br /><div><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div><br /></div></div>shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-48754999098931126582020-09-29T03:58:00.013+05:302020-09-29T05:03:02.997+05:30Ending Song: Meghe Dhaka Tara (2013)<p>Meghe Dhaka Tara (2013) directed by Kamaleshwar Mukherjee</p><p>Have not seen a better or a more intensely rousing movie ending scene than this one. Absolutely electrifying! The song is also just marvelous, riveting. The landscape and the scenes of the video song are just brilliant! Masterpiece. Every time I see or hear it, I get goosebumps. <a href="https://www.youtube.com/watch?v=1LFMS3GEiTw" target="_blank">Watch the video: Aaohaan</a>.</p><p>Meghe Dhaka Tara (Cloud Capped Star) is a 2013 Indian Bengali film directed by Kamaleswar Mukherjee. The film is inspired from the life and works of Bengali film director Ritwik Ghatak. The entire film is in black and white except the last scene which has been shot in colour. In this film Saswata Chatterjee plays the character of Nilkantha Bagchi and Ananya Chatterjee plays the role of Durga, Nilkantha's wife. Besides giving an account of Ghatak's life, the film also depicts the socio-political environment of contemporary West Bengal during the Tebhaga and Naxalite movements.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://www.youtube.com/watch?v=1LFMS3GEiTw" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" data-original-height="573" data-original-width="1154" height="354" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAyqqurr9xJMtjd3On2o272eYBzKWno3Xx0KZmmUU1xcDJN63JSnOpFViBh7VO2987t-2UdwUfUXkxV4euXVdBoMpVcI3tt-VUToe1NL0UbAWabc3Wta2RIj-VkmOQM3IXQ_8m4w/w712-h354/MDT.png" width="712" /></a></div><br />Kamaleshwar Mukherjee has used the metaphor of Bangabala in the closing shots of his film, where a young girl, Bangabala, from Ghatak's Jukti Takko Aar Golpo is given concrete shape to look like the rape victim Phoolmoni in the mental home (in the 2013 film). She waits for Neelkantha out in the fields, as if inviting him to join her back, to his roots in Pabna and other places in East Pakistan once a part of undivided Bengal. Mukherjee beautifully suggests the 'liberation' of Ghatak who frees himself to join his 'Bangabala' to go on a new journey and perhaps begin to rewrite his life again. Phoolmoni is liberated and so is Neelkantha. The director does not use the darkness of death but the brightness of a yellow horizon in the distance, shown in bright colour, where the two figures become smaller and smaller till the film freezes to a close.<p></p><p>Background:</p><p>Ghatak’s last film featured himself as the drunken and spent intellectual Neelkantha who goes on a picaresque journey through Bengal to reconcile himself with his wife. He is accompanied by Nachiketa (Burman) and Bangabala (S. Mitra), a young refugee from Bangladesh. On the way they are joined by a Sanskrit teacher, Jagannath (Bhattacharya). The episodic narrative also includes encounters with Shatrujit (Dutt) who was once a noted writer but who now writes pornography (apparently a reference to novelist Samaresh Bose); a ranting trade union leader and Panchanan Ustad (Mukherjee) who makes masks for Chhou dancers (a sequence is devoted to showing the famous dance). Jagannath is shot by a landlord when the group stumbles upon a land-grab action. The film ends with Neelkantha meeting a group of Naxalite students wanted by the police: he argues politics with them and is shot in a police ambush the next morning. </p><p>Filmed while Ghatak was ill and suffering from alcoholism shortly before his death, Jukti is an inventive and lucid though pessimistic testament film, acted with elegance and irony by the director. With an astonishing sense of freedom Ghatak weaves together different styles and images ranging from gross calendar art (the courtship of his wife) to an almost abstract dance of death; from the elaborate Chhou performance where the goddess Durga slays the demon to lyrical depictions of nature; from inserted bits of leader footage to a Baul song. The encounters with the pornographer and the Naxalites add up to a devastating critique of contemporary politics. In the end, Ghatak offers a disabused but stubborn politics of the everyday: Neelkantha dies with a quote from the Manik Bandyopadhyay story Shilpi about a weaver who wove an empty loom because ‘one must do something’. Geeta Kapur’s essay ‘Articulating the Self into History’ (1989) is the most extended study on the film.</p><p><b>Ritwik Ghatak was an epitome of 'anarchy in art', especially film-making! <a href="https://www.britannica.com/topic/anarchism/Anarchism-in-the-arts" target="_blank">Anarchism in the Arts (Britannica link)</a></b></p>shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-32316632059564450902019-05-07T03:21:00.001+05:302019-05-07T03:31:02.613+05:30Trial of SPARQL on GraphDB<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://www.w3.org/Icons/SW/sw-horz-w3c.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="48" data-original-width="241" src="https://www.w3.org/Icons/SW/sw-horz-w3c.png" /></a></div>
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I used the <a href="https://www.w3.org/OWL/" target="_blank">OWL</a> ontology files from <a href="https://bioportal.bioontology.org/ontologies/ABA-AMB" style="box-sizing: border-box; color: #343a40; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 0px; padding: 0px; text-decoration-line: none;">Allen Brain Atlas (ABA) Adult Mouse Brain Ontology</a></div>
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Then used <a href="https://www.w3.org/TR/rdf-sparql-query/#introduction" target="_blank">SPARQL</a> Query on <a href="http://graphdb.ontotext.com/documentation/standard/introduction-to-semantic-web.html#ontologies-and-knowledge-bases" target="_blank">GraphDB</a><br />
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<a href="http://localhost:7200/import#user">http://localhost:7200/import#user</a><br />
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First you need to Import the file using Import <span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: x-small;">→ </span>RDF <span style="background-color: white; color: #222222; font-family: "arial" , sans-serif; font-size: x-small;">→</span> Server Files</div>
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Then use SPARQL Query & Update - http://localhost:7200/sparql<br />
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Sample SPARQL query that I used:</div>
<blockquote class="tr_bq" style="text-align: left;">
<span style="color: blue; font-family: "courier new" , "courier" , monospace;">select distinct ?o where {</span><br />
<span style="color: blue; font-family: "courier new" , "courier" , monospace;"> ?s ?p ?o .</span><br />
<span style="color: blue; font-family: "courier new" , "courier" , monospace;"> filter contains(?o,"Cerebral cortex") #-- the important line</span><br />
<span style="color: blue; font-family: "courier new" , "courier" , monospace;">}</span><br />
<span style="color: blue; font-family: "courier new" , "courier" , monospace;">limit 100</span></blockquote>
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Result:</div>
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<table border="0" cellpadding="0" cellspacing="0" class="resultsTable table stripe hover table-bordered fixedCellWidth dataTable no-footer JColResizer" id="DataTables_Table_2" role="grid" style="background-color: white; border-collapse: collapse; border-color: rgb(236, 238, 239) rgb(236, 238, 239) rgb(17, 17, 17); border-image: initial; border-spacing: 0px; border-style: solid; border-width: 1px; box-sizing: content-box; clear: both; color: #373a3c; font-family: RubikWeb, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif; font-size: 14px; font-variant-ligatures: no-common-ligatures; margin: 0px auto; max-width: 100%; overflow-wrap: break-word; table-layout: auto; width: 1082px;"><tbody style="box-sizing: inherit;">
<tr class="odd" role="row" style="background-color: #f9f9f9; box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">1</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 5</div>
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</td></tr>
<tr class="even" role="row" style="box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">2</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex</div>
</div>
</td></tr>
<tr class="odd" role="row" style="background-color: #f9f9f9; box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">3</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 2-3</div>
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</td></tr>
<tr class="even" role="row" style="box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">4</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 6a</div>
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</td></tr>
<tr class="odd" role="row" style="background-color: #f9f9f9; box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">5</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layers 1-6b</div>
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</td></tr>
<tr class="even" role="row" style="box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">6</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 1</div>
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</td></tr>
<tr class="odd" role="row" style="background-color: #f9f9f9; box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">7</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 6</div>
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</td></tr>
<tr class="even" role="row" style="box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">8</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 3</div>
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</td></tr>
<tr class="odd" role="row" style="background-color: #f9f9f9; box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">9</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 2</div>
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</td></tr>
<tr class="even" role="row" style="background-color: whitesmoke; box-sizing: inherit;"><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; color: #999999; max-width: 50vw; overflow: hidden; padding: 8px 10px; text-align: center; vertical-align: top;">10</td><td style="border: 1px solid rgb(221, 221, 221); box-sizing: content-box; max-width: 50vw; overflow: hidden; padding: 8px 10px; vertical-align: top;"><div style="box-sizing: inherit; padding: 0px 5px; word-break: break-all;">
<div class="nonUri" style="background-color: transparent; border: none; box-sizing: inherit; font-family: inherit; font-size: inherit; padding: 0px; white-space: pre-wrap;">
Cerebral cortex, layer 4</div>
</div>
</td></tr>
</tbody></table>
</div>
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Schema.org - has more RDF / triples: <a href="https://schema.org/docs/developers.html">https://schema.org/docs/developers.html</a></div>
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Another useful link: <a href="https://neo4j.com/graphconnect-2018/session/ontologies-in-neo4j-semantics-knowledge-graphs" target="_blank">Ontologies in Neo4j: Semantics and Knowledge Graphs</a> </div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-85544210262065543262019-01-12T02:24:00.001+05:302019-01-12T03:34:29.787+05:30Run MySQL without Administrator privileges<div dir="ltr" style="text-align: left;" trbidi="on">
Create a text file containing the password-assignment statement on a single line. Replace the password with the password that you want to use.<br />
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<span style="font-family: "courier new" , "courier" , monospace;">ALTER USER 'root'@'localhost' IDENTIFIED BY 'MyNewPass';</span><br />
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Save the file. This example assumes that you name the file <span style="font-family: Courier New, Courier, monospace;">C:\mysql-init.txt</span><br />
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Open a console window to get to the command prompt: From the Start menu, select Run, then enter cmd as the command to be run.<br />
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Start the MySQL server with the special <span style="font-family: Courier New, Courier, monospace;">--init-file</span> option (notice that the backslash in the option value is doubled):<br />
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<span style="font-family: "courier new" , "courier" , monospace;">cd "C:\Program Files\MySQL\MySQL Server 8.0\bin"</span><br />
<span style="font-family: "courier new" , "courier" , monospace;">mysqld --init-file=C:\\mysql-init.txt</span><br />
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-83791146835961116292018-01-05T07:05:00.001+05:302018-01-05T08:40:48.721+05:30Softmax function, softmax regression.<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="color: #222222; font-family: sans-serif;"><span style="font-size: 14px;"><span style="background-color: white;">T</span>he softmax function is also called the normalized exponential function. </span></span><br />
<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">It is a generalization of the <a href="https://en.wikipedia.org/wiki/Logistic_function" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #0b0080; text-decoration-line: none;" title="Logistic function">logistic function</a> that "squashes" a <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 16.52px; font-style: italic; line-height: 1; white-space: nowrap;">K</span>-dimensional vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation></semantics></math></span><img alt="\mathbf {z} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82eca5d0928078d5a61b9e7e98cc73db31070909" style="border: 0px; display: inline-block; height: 1.676ex; margin: 0px; vertical-align: -0.338ex; width: 1.198ex;" /></span> of arbitrary real values to a <span class="texhtml mvar" style="font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 16.52px; font-style: italic; line-height: 1; white-space: nowrap;">K</span>-dimensional vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px 1px 1px 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \sigma (\mathbf {z} )}</annotation></semantics></math></span><img alt="\sigma (\mathbf {z} )" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e610a6185b8850a6f567c4902387b17f0ec1652" style="border: 0px; display: inline-block; height: 2.843ex; margin: 0px; vertical-align: -0.838ex; width: 4.369ex;" /></span> of real values in the range [0, 1] that add up to 1. </span><br />
<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">In </span><a href="https://en.wikipedia.org/wiki/Probability_theory" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Probability theory">probability theory</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, </span><span style="color: #222222; font-family: sans-serif;"><span style="font-size: 14px;">t</span></span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">he output of the softmax function can be used to represent a </span><a href="https://en.wikipedia.org/wiki/Categorical_distribution" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Categorical distribution">categorical distribution</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> – that is, a </span><a href="https://en.wikipedia.org/wiki/Probability_distribution" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Probability distribution">probability distribution</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> over </span><span class="texhtml mvar" style="background-color: white; color: #222222; font-family: "nimbus roman no9 l" , "times new roman" , "times" , serif; font-size: 16.52px; font-style: italic; line-height: 1; white-space: nowrap;">K</span><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> different possible outcomes. </span><br />
<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"><br /></span>
<span style="background-color: white; color: blue; font-family: sans-serif; font-size: 14px;">Example: </span><br />
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<span style="font-family: "roboto" , sans-serif; font-size: 16px;">We know that every image in MNIST is of a handwritten digit between zero and nine. So there are only ten possible things that a given image can be. We want to be able to look at an image and give the probabilities for it being each digit. For example, our model might look at a picture of a nine and be 85% sure it's a nine, but give a 5% chance to it being an eight (because of the top loop) and a bit of probability to all the others because it isn't 100% sure.</span></span><br />
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<span style="color: blue;">This is a classic case where a softmax regression is a natural, simple model. If you want to assign probabilities to an object being one of several different things, softmax is the thing to do, because softmax gives us a list of values between 0 and 1 that add up to 1. Even to train more sophisticated models, the final step is usually a layer of softmax.</span></div>
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<b>A softmax regression has two steps: first we add up the evidence of our input being in certain classes, and then we convert that evidence into probabilities.</b></div>
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To tally up the evidence that a given image is in a particular class, we do a weighted sum of the pixel intensities. The weight is negative if that pixel having a high intensity is evidence against the image being in that class, and positive if it is evidence in favor.</div>
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We also add some extra evidence called a bias. Basically, we want to be able to say that some things are more likely independent of the input. The result is that the evidence for a class <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" id="MathJax-Element-4-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.903ex" role="img" style="vertical-align: -0.241ex;" viewbox="0 -715.3 345.5 819.3" width="0.802ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-69" y="0"><path d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" id="MJMATHI-69" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">i</math></span></span>given an input <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" id="MathJax-Element-5-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.429ex" role="img" style="vertical-align: -0.241ex;" viewbox="0 -511.5 572.5 615.4" width="1.33ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-78" y="0"><path d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" id="MJMATHI-78" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">x</math></span></span> is:</div>
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where <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>i</mi></msub></math>" id="MathJax-Element-7-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="2.376ex" role="img" style="vertical-align: -0.596ex;" viewbox="0 -766.3 1288.8 1023.1" width="2.993ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-57" y="0"><path d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z" id="MJMATHI-57" stroke-width="1"></path></use><use transform="scale(0.707)" x="1335" xlink:href="#MJMATHI-69" y="-213"><path d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" id="MJMATHI-69" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>i</mi></msub></math></span></span> is the weights and <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math>" id="MathJax-Element-8-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="2.376ex" role="img" style="vertical-align: -0.596ex;" viewbox="0 -766.3 773.8 1023.1" width="1.797ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-62" y="0"><path d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z" id="MJMATHI-62" stroke-width="1"></path></use><use transform="scale(0.707)" x="607" xlink:href="#MJMATHI-69" y="-213"><path d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" id="MJMATHI-69" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math></span></span> is the bias for class <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" id="MathJax-Element-9-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.903ex" role="img" style="vertical-align: -0.241ex;" viewbox="0 -715.3 345.5 819.3" width="0.802ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-69" y="0"><path d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" id="MJMATHI-69" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">i</math></span></span>, and <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" id="MathJax-Element-10-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="2.376ex" role="img" style="margin-left: -0.027ex; vertical-align: -0.715ex;" viewbox="-11.5 -715.3 424 1023.1" width="0.985ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-6A" y="0"><path d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" id="MJMATHI-6A" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">j</math></span></span> is an index for summing over the pixels in our input image <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" id="MathJax-Element-11-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.429ex" role="img" style="vertical-align: -0.241ex;" viewbox="0 -511.5 572.5 615.4" width="1.33ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-78" y="0"><path d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" id="MJMATHI-78" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">x</math></span></span>. We then convert the evidence tallies into our predicted probabilities <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" id="MathJax-Element-12-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.903ex" role="img" style="vertical-align: -0.715ex;" viewbox="0 -511.5 497.5 819.3" width="1.155ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-79" y="0"><path d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" id="MJMATHI-79" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">y</math></span></span> using the "softmax" function:</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 0px; padding: 0px;">
<span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><br />
<div class="MathJax_SVG_Display" style="box-sizing: inherit; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-x: auto; padding: 0px; position: relative; text-align: center; width: 805px;">
<span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>y</mi><mo>=</mo><mtext>softmax</mtext><mo stretchy="false">(</mo><mtext>evidence</mtext><mo stretchy="false">)</mo></math>" id="MathJax-Element-13-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="2.613ex" role="img" style="vertical-align: -0.715ex;" viewbox="0 -817.3 9762.1 1125" width="22.673ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-79" y="0"><path d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" id="MJMATHI-79" stroke-width="1"></path></use><use x="775" xlink:href="#MJMAIN-3D" y="0"><path d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" id="MJMAIN-3D" stroke-width="1"></path></use><g transform="translate(1831,0)"><use xlink:href="#MJMAIN-73"><path d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" id="MJMAIN-73" stroke-width="1"></path></use><use x="394" xlink:href="#MJMAIN-6F" y="0"><path d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z" id="MJMAIN-6F" stroke-width="1"></path></use><use x="895" xlink:href="#MJMAIN-66" y="0"><path d="M273 0Q255 3 146 3Q43 3 34 0H26V46H42Q70 46 91 49Q99 52 103 60Q104 62 104 224V385H33V431H104V497L105 564L107 574Q126 639 171 668T266 704Q267 704 275 704T289 705Q330 702 351 679T372 627Q372 604 358 590T321 576T284 590T270 627Q270 647 288 667H284Q280 668 273 668Q245 668 223 647T189 592Q183 572 182 497V431H293V385H185V225Q185 63 186 61T189 57T194 54T199 51T206 49T213 48T222 47T231 47T241 46T251 46H282V0H273Z" id="MJMAIN-66" stroke-width="1"></path></use><use x="1201" xlink:href="#MJMAIN-74" y="0"><path d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z" id="MJMAIN-74" stroke-width="1"></path></use><use x="1591" xlink:href="#MJMAIN-6D" y="0"><path d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 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</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
Here softmax is serving as an "activation" or "link" function, shaping the output of our linear function into the form we want -- in this case, a probability distribution over 10 cases. You can think of it as converting tallies of evidence into probabilities of our input being in each class. It's defined as:</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 0px; padding: 0px;">
<span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><br />
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-249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" id="MJMAIN-29" stroke-width="1"></path></use><use x="20070" xlink:href="#MJMAIN-29" y="0"><path d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" id="MJMAIN-29" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 401.672px;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>softmax</mtext><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>normalize</mtext><mo stretchy="false">(</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></div>
</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
If you expand that equation out, you get:</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 0px; padding: 0px;">
<span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><br />
<div class="MathJax_SVG_Display" style="box-sizing: inherit; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-x: auto; padding: 0px; position: relative; text-align: center; width: 805px;">
<span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>softmax</mtext><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><msub><mo stretchy="false">)</mo><mi>i</mi></msub><mo>=</mo><mfrac><mrow><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><msub><mi>e</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><munder><mo>∑</mo><mi>j</mi></munder><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><msub><mi>e</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></math>" id="MathJax-Element-15-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; 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stretchy="false">)</mo><mi>i</mi></msub><mo>=</mo><mfrac><mrow><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><msub><mi>e</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><munder><mo>∑</mo><mi>j</mi></munder><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>c</mi><msub><mi>e</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></math></span></span></div>
</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
But it's often more helpful to think of softmax the first way: exponentiating its inputs and then normalizing them. The exponentiation means that one more unit of evidence increases the weight given to any hypothesis multiplicatively. And conversely, having one less unit of evidence means that a hypothesis gets a fraction of its earlier weight. No hypothesis ever has zero or negative weight. Softmax then normalizes these weights, so that they add up to one, forming a valid probability distribution. (To get more intuition about the softmax function, check out the <a href="http://neuralnetworksanddeeplearning.com/chap3.html#softmax" style="box-sizing: inherit; color: #039be5; outline: 0px; text-decoration-line: none;">section</a> on it in Michael Nielsen's book, complete with an interactive visualization.)</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
You can picture our softmax regression as looking something like the following, although with a lot more <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" id="MathJax-Element-16-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.429ex" role="img" style="vertical-align: -0.241ex;" viewbox="0 -511.5 572.5 615.4" width="1.33ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-78" y="0"><path d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" id="MJMATHI-78" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">x</math></span></span>s. For each output, we compute a weighted sum of the <span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><span class="MathJax_SVG" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" id="MathJax-Element-17-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline-block; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><svg aria-hidden="true" focusable="false" height="1.429ex" role="img" style="vertical-align: -0.241ex;" viewbox="0 -511.5 572.5 615.4" width="1.33ex" xmlns:xlink="http://www.w3.org/1999/xlink"><g fill="currentColor" stroke-width="0" stroke="currentColor" transform="matrix(1 0 0 -1 0 0)"><use x="0" xlink:href="#MJMATHI-78" y="0"><path d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" id="MJMATHI-78" stroke-width="1"></path></use></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML">x</math></span></span>s, add a bias, and then apply softmax.</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 20px auto 10px; padding: 0px; width: 442.75px;">
<img src="https://www.tensorflow.org/images/softmax-regression-scalargraph.png" style="border: 0px; box-sizing: inherit; margin: 0px; max-width: 100%; padding: 0px; width: 442.75px;" /></div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
If we write that out as equations, we get:</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 20px 0px 10px 201.25px; padding: 0px; width: 418.594px;">
<img alt="[y1, y2, y3] = softmax(W11*x1 + W12*x2 + W13*x3 + b1, W21*x1 + W22*x2 + W23*x3 + b2, W31*x1 + W32*x2 + W33*x3 + b3)" src="https://www.tensorflow.org/images/softmax-regression-scalarequation.png" style="border: 0px; box-sizing: inherit; margin: 0px; max-width: 100%; padding: 0px; width: 418.594px;" /></div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
We can "vectorize" this procedure, turning it into a matrix multiplication and vector addition. This is helpful for computational efficiency. (It's also a useful way to think.)</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 20px auto 10px; padding: 0px; width: 402.5px;">
<img alt="[y1, y2, y3] = softmax([[W11, W12, W13], [W21, W22, W23], [W31, W32, W33]]*[x1, x2, x3] + [b1, b2, b3])" src="https://www.tensorflow.org/images/softmax-regression-vectorequation.png" style="border: 0px; box-sizing: inherit; margin: 0px; max-width: 100%; padding: 0px; width: 402.5px;" /></div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
More compactly, we can just write:</div>
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin: 0px; padding: 0px;">
<span class="MathJax_Preview" style="box-sizing: inherit; color: inherit;"></span><br />
<div class="MathJax_SVG_Display" style="box-sizing: inherit; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-x: auto; padding: 0px; position: relative; text-align: center; width: 805px;">
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442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" id="MJMATHI-79" stroke-width="1"></path></use><use x="775" xlink:href="#MJMAIN-3D" y="0"><path d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" id="MJMAIN-3D" 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class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="border: 0px !important; box-sizing: inherit; clip: rect(1px 1px 1px 1px); display: block !important; height: 1px !important; left: 0px; overflow: hidden !important; padding: 1px 0px 0px !important; position: absolute !important; top: 0px; transition: none; user-select: none; width: 183.313px;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>softmax</mtext><mo stretchy="false">(</mo><mi>W</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></math></span></span></div>
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<span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;">The idea of softmax is to define a new type of output layer for our neural networks. It begins in the same way as with a sigmoid layer, by forming the weighted inputs*</span><span class="marginnote" style="background-color: white; color: #666666; display: inline-block; font-family: "georgia" , "times new roman" , serif; font-size: 9pt; left: 680px; margin: 0px; padding: 0px; position: absolute; width: 260px;"><br /></span><span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;"> </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>z</mi><mi>j</mi><mi>L</mi></msubsup><mo>=</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></munder><msubsup><mi>w</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mi>k</mi></mrow><mi>L</mi></msubsup><msubsup><mi>a</mi><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><msubsup><mi>b</mi><mi>j</mi><mi>L</mi></msubsup></math>" id="MathJax-Element-183-Frame" role="presentation" style="background-color: white; 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display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2576" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.816em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2577" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.385em 1001.33em 4.267em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.458em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2578" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2579" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2580" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-2581" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-2582" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.385em 1000.39em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2583" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">k<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-2584" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-2585" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.248em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.189em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2586" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">b</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2587" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2588" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.502em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.497em; width: 0px;"></span></span></nobr></span><span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;"> However, we don't apply the sigmoid function to get the output. Instead, in a softmax layer we apply the so-called </span><em style="background-color: white; color: #333333; font-family: Georgia, "Times New Roman", serif; font-size: 20px;">softmax function</em><span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;"> to the </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>z</mi><mi>j</mi><mi>L</mi></msubsup></math>" id="MathJax-Element-184-Frame" role="presentation" style="background-color: white; border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: "georgia" , "times new roman" , serif; font-size: 20px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2589" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.934em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.885em;"><span style="border: 0px; clip: rect(1.179em 1000.88em 2.699em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.105em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2590" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2591" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.885em;"><span style="border: 0px; clip: rect(3.434em 1000.39em 4.267em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2592" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">z</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.395em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2593" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.395em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2594" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.11em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.353em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.497em; width: 0px;"></span></span></nobr></span><span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;">. According to this function, the activation </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mi>j</mi><mi>L</mi></msubsup></math>" id="MathJax-Element-185-Frame" role="presentation" style="background-color: white; border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: "georgia" , "times new roman" , serif; font-size: 20px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2595" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 1.032em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(1.179em 1000.98em 2.699em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.105em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2596" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2597" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2598" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2599" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2600" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.11em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.353em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.497em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; clip: rect(1px 1px 1px 1px); display: inline; height: 1px !important; left: 0px; line-height: normal; margin: 0px; overflow: hidden !important; padding: 0px; position: static; top: 0px; transition: none; user-select: none; vertical-align: 0px; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mi>j</mi><mi>L</mi></msubsup></math></span></span><span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;"> of the </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" id="MathJax-Element-186-Frame" role="presentation" style="background-color: white; border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: "georgia" , "times new roman" , serif; font-size: 20px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2601" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.297em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.297em;"><span style="border: 0px; clip: rect(1.718em 1000.3em 2.846em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.498em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2602" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2603" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.002em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.247em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; clip: rect(1px 1px 1px 1px); display: inline; height: 1px !important; left: 0px; line-height: normal; margin: 0px; overflow: hidden !important; padding: 0px; position: static; top: 0px; transition: none; user-select: none; vertical-align: 0px; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></span></span><span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;">th output neuron is</span><a class="displaced_anchor" href="https://www.blogger.com/null" name="eqtn78" style="background-color: white; color: #2a6ea6; font-family: Georgia, "Times New Roman", serif; font-size: 20px;"></a><br />
<div class="MathJax_Display" style="background-color: white; color: #333333; font-family: Georgia, "Times New Roman", serif; font-size: 20px; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; position: relative; text-align: center; width: 600px;">
<span class="MathJax MathJax_FullWidth" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-78"><mtext>(78)</mtext></mtd><mtd><msubsup><mi>a</mi><mi>j</mi><mi>L</mi></msubsup><mo>=</mo><mfrac><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><msubsup><mi>z</mi><mi>j</mi><mi>L</mi></msubsup></mrow></msup><mrow><munder><mo>∑</mo><mi>k</mi></munder><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><msubsup><mi>z</mi><mi>k</mi><mi>L</mi></msubsup></mrow></msup></mrow></mfrac><mo>,</mo></mtd></mlabeledtr></mtable></math>" id="MathJax-Element-187-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: table-cell; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; width: 10000em; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2604" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; min-width: 10.64em; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 600px;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; min-width: 10.64em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 600px;"><span style="border: 0px; clip: rect(2.208em 1005.44em 5.346em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px; width: 600px;"><span class="mrow" id="MathJax-Span-2605" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mtable" id="MathJax-Span-2606" style="border: 0px; display: inline; line-height: normal; margin: 0px; min-width: 10.64em; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; min-width: 10.64em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 600px;"><span style="border: 0px; clip: rect(-1.811em 1005.44em 1.326em -999.998em); display: inline-block; height: 0px; left: 300px; line-height: normal; margin: 0px 0px 0px -2.743em; padding: 0px; position: absolute; top: 0em; transition: none; vertical-align: 0px; width: 5.493em;"><span style="border: 0px; clip: rect(2.208em 1005.44em 5.346em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 5.493em;"><span style="border: 0px; clip: rect(2.208em 1005.44em 5.346em -999.998em); line-height: normal; margin: 0px; padding: 0px; position: absolute; right: 0em; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mtd" id="MathJax-Span-2610" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2611" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2612" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2613" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2614" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2615" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-2616" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.297em; position: static; transition: none; vertical-align: 0px;">=</span><span class="mfrac" id="MathJax-Span-2617" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.297em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.1em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.748em;"><span style="border: 0px; clip: rect(2.895em 1001.13em 4.169em -999.998em); left: 28.0156px; line-height: normal; margin: 0px 0px 0px -0.537em; padding: 0px; position: absolute; top: -4.703em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2618" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.13em;"><span style="border: 0px; clip: rect(3.434em 1000.39em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2619" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.444em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.458em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2620" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2621" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2622" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.591em;"><span style="border: 0px; clip: rect(3.581em 1000.25em 4.218em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2623" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">z</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.35em 4.169em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.262em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2624" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.2em 4.267em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.821em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2625" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(2.944em 1002.65em 4.463em -999.998em); left: 28.0156px; line-height: normal; margin: 0px 0px 0px -1.321em; padding: 0px; position: absolute; top: -3.135em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2626" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="munderover" id="MathJax-Span-2627" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.326em;"><span style="border: 0px; clip: rect(3.091em 1000.84em 4.414em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2628" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.002em;">∑</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.934em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2629" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">k<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2630" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.199em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.13em;"><span style="border: 0px; clip: rect(3.434em 1000.39em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2631" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.444em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2632" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2633" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2634" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.591em;"><span style="border: 0px; clip: rect(3.581em 1000.25em 4.218em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2635" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">z</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.35em 4.169em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.262em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2636" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.3em 4.169em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.772em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2637" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">k<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.885em 1002.75em 1.228em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.272em; transition: none; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0em; width: 2.748em;"></span><span style="border: 0px; display: inline-block; height: 1.081em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-2638" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">,</span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; clip: rect(-0.831em 1001.62em 0.346em -999.998em); display: inline-block; height: 0px; line-height: normal; margin: 0px 0em 0px 0px; padding: 0px; position: absolute; right: 0em; top: 0em; transition: none; vertical-align: 0px; width: 1.669em;"><span style="border: 0px; clip: rect(3.189em 1001.62em 4.365em -999.998em); line-height: normal; margin: 0px; padding: 0px; position: absolute; right: 0em; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mtd" id="mjx-eqn-78" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2608" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mtext" id="MathJax-Span-2609" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">(78)</span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 3.002em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.248em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="border: 0px; clip: rect(1px 1px 1px 1px); display: inline; height: 1px !important; left: 0px; line-height: normal; margin: 0px; overflow: hidden !important; padding: 0px; position: static; top: 0px; transition: none; user-select: none; vertical-align: 0px; width: 600px;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"></math><br /></span></span></div>
<span style="background-color: white; color: #333333; font-family: "georgia" , "times new roman" , serif; font-size: 20px;">where in the denominator we sum over all the output neurons.</span><br />
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<div style="background-color: white; color: #333333; font-family: Georgia, "Times New Roman", serif; font-size: 20px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.7;">
In fact, if you look closely, you'll see that in both cases the total change in the other activations exactly compensates for the change in <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mn>4</mn><mi>L</mi></msubsup></math>" id="MathJax-Element-205-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2806" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 1.032em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(1.179em 1000.98em 2.551em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.105em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2807" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2808" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2809" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2810" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.385em 1000.44em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2811" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">4</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.11em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.252em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span>. The reason is that the output activations are guaranteed to always sum up to <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" id="MathJax-Element-206-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2812" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.493em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.493em;"><span style="border: 0px; clip: rect(1.669em 1000.39em 2.65em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.498em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2813" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2814" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 0.802em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.048em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; clip: rect(1px 1px 1px 1px); display: inline; height: 1px !important; left: 0px; line-height: normal; margin: 0px; overflow: hidden !important; padding: 0px; position: static; top: 0px; transition: none; user-select: none; vertical-align: 0px; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span>, as we can prove using Equation <span class="equation_link" id="margin_321011740331_reveal" style="color: #2a6ea6;">(78)</span> and a little algebra:<a class="displaced_anchor" href="https://www.blogger.com/null" name="eqtn79" style="color: #2a6ea6;"></a></div>
<div class="MathJax_Display" style="color: rgb(51, 51, 51) !important; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; position: relative; text-align: center; width: 600px;">
<span class="MathJax MathJax_FullWidth" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-79"><mtext>(79)</mtext></mtd><mtd><munder><mo>∑</mo><mi>j</mi></munder><msubsup><mi>a</mi><mi>j</mi><mi>L</mi></msubsup></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><munder><mo>∑</mo><mi>j</mi></munder><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><msubsup><mi>z</mi><mi>j</mi><mi>L</mi></msubsup></mrow></msup></mrow><mrow><munder><mo>∑</mo><mi>k</mi></munder><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><msubsup><mi>z</mi><mi>k</mi><mi>L</mi></msubsup></mrow></msup></mrow></mfrac><mo>=</mo><mn>1.</mn></mtd></mlabeledtr></mtable></math>" id="MathJax-Element-208-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: table-cell; float: none; font-size: 20px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; width: 10000em; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2846" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; min-width: 13.924em; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 600px;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; min-width: 13.924em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 600px;"><span style="border: 0px; clip: rect(2.257em 1008.73em 5.836em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.311em; transition: none; vertical-align: 0px; width: 600px;"><span class="mrow" id="MathJax-Span-2847" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mtable" id="MathJax-Span-2848" style="border: 0px; display: inline; line-height: normal; margin: 0px; min-width: 13.924em; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; min-width: 13.924em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 600px;"><span style="border: 0px; clip: rect(-2.056em 1008.73em 1.522em -999.998em); display: inline-block; height: 0px; left: 300px; line-height: normal; margin: 0px 0px 0px -4.409em; padding: 0px; position: absolute; top: 0em; transition: none; vertical-align: 0px; width: 8.777em;"><span style="border: 0px; clip: rect(2.895em 1002.45em 5.542em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.453em;"><span style="border: 0px; clip: rect(2.895em 1002.45em 5.542em -999.998em); line-height: normal; margin: 0px; padding: 0px; position: absolute; right: 0em; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mtd" id="MathJax-Span-2852" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2853" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="munderover" id="MathJax-Span-2854" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.277em;"><span style="border: 0px; clip: rect(2.895em 1001.23em 4.659em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2855" style="border: 0px; display: inline; font-family: "stixsizeonesym"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: -0.537em;">∑</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.2em 4.414em -999.998em); left: 0.542em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.89em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2856" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2857" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.199em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2858" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2859" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2860" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.483em 1000.93em 4.169em -999.998em); left: 2.453em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.483em 1000.93em 4.169em -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.488em; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mtd" id="MathJax-Span-2861" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2862" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2863" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"></span><span class="mo" id="MathJax-Span-2864" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.297em; position: static; transition: none; vertical-align: 0px;">=</span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(2.012em 1004.95em 5.395em -999.998em); left: 3.777em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.066em; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 5.051em;"><span style="border: 0px; clip: rect(2.012em 1004.95em 5.395em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.066em; transition: none; vertical-align: 0px;"><span class="mtd" id="MathJax-Span-2865" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2866" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2867" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.1em; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.748em;"><span style="border: 0px; clip: rect(2.895em 1002.5em 4.61em -999.998em); left: 28.0156px; line-height: normal; margin: 0px 0px 0px -1.272em; padding: 0px; position: absolute; top: -4.9em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2868" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="munderover" id="MathJax-Span-2869" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.179em;"><span style="border: 0px; clip: rect(3.091em 1000.84em 4.414em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2870" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.002em;">∑</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.934em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2871" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2872" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.199em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.13em;"><span style="border: 0px; clip: rect(3.434em 1000.39em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2873" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.444em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.458em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2874" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2875" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2876" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.591em;"><span style="border: 0px; clip: rect(3.581em 1000.25em 4.218em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2877" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">z</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.35em 4.169em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.262em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2878" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.2em 4.267em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.821em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2879" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(2.944em 1002.65em 4.463em -999.998em); left: 28.0156px; line-height: normal; margin: 0px 0px 0px -1.321em; padding: 0px; position: absolute; top: -3.135em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2880" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="munderover" id="MathJax-Span-2881" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.326em;"><span style="border: 0px; clip: rect(3.091em 1000.84em 4.414em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2882" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.002em;">∑</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.934em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2883" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">k<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2884" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.199em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.13em;"><span style="border: 0px; clip: rect(3.434em 1000.39em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2885" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">e</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.444em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2886" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2887" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2888" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.591em;"><span style="border: 0px; clip: rect(3.581em 1000.25em 4.218em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2889" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">z</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.35em 4.169em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.262em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2890" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.532em 1000.3em 4.169em -999.998em); left: 0.297em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.772em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2891" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 10.2px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">k<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.885em 1002.75em 1.228em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.272em; transition: none; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0em; width: 2.748em;"></span><span style="border: 0px; display: inline-block; height: 1.081em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-2892" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.297em; position: static; transition: none; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-2893" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.297em; position: static; transition: none; vertical-align: 0px;">1.</span></span></span><span style="border: 0px; display: inline-block; height: 4.071em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; display: inline-block; height: 4.071em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border: 0px; clip: rect(-0.831em 1001.62em 0.346em -999.998em); display: inline-block; height: 0px; line-height: normal; margin: 0px 0em 0px 0px; padding: 0px; position: absolute; right: 0em; top: 0em; transition: none; vertical-align: 0px; width: 1.669em;"><span style="border: 0px; clip: rect(3.189em 1001.62em 4.365em -999.998em); line-height: normal; margin: 0px; padding: 0px; position: absolute; right: 0em; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mtd" id="mjx-eqn-79" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2850" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mtext" id="MathJax-Span-2851" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">(79)</span></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 4.316em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 3.452em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.448em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="border: 0px; clip: rect(1px 1px 1px 1px); display: inline; height: 1px !important; left: 0px; line-height: normal; margin: 0px; overflow: hidden !important; padding: 0px; position: static; top: 0px; transition: none; user-select: none; vertical-align: 0px; width: 600px;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"></math><br /></span></span></div>
As a result, if <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mn>4</mn><mi>L</mi></msubsup></math>" id="MathJax-Element-209-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-size: 20px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2894" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 1.032em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(1.179em 1000.98em 2.551em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.105em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2895" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2896" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2897" style="border: 0px; display: inline; font-family: "stixgeneral"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2898" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.385em 1000.44em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2899" style="border: 0px; display: inline; font-family: "stixgeneral"; font-size: 14.4228px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">4</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.11em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.252em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> increases, then the other output activations must decrease by the same total amount, to ensure the sum over all activations remains <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" id="MathJax-Element-210-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-size: 20px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2900" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.493em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.493em;"><span style="border: 0px; clip: rect(1.669em 1000.39em 2.65em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.498em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2901" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2902" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 0.802em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.048em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; clip: rect(1px 1px 1px 1px); display: inline; height: 1px !important; left: 0px; line-height: normal; margin: 0px; overflow: hidden !important; padding: 0px; position: static; top: 0px; transition: none; user-select: none; vertical-align: 0px; width: 1px !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span>. And, of course, similar statements hold for all the other activations.<br />
<br />
<div style="background-color: white; color: #333333; font-family: Georgia, "Times New Roman", serif; font-size: 20px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.7;">
Equation <span class="equation_link" id="margin_461714771189_reveal" style="color: #2a6ea6;">(78)</span> also implies that the output activations are all positive, since the exponential function is positive. Combining this with the observation in the last paragraph, we see that the output from the softmax layer is a set of positive numbers which sum up to <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" id="MathJax-Element-212-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2934" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.493em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.493em;"><span style="border: 0px; clip: rect(1.669em 1000.39em 2.65em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.498em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2935" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2936" style="border: 0px; display: inline; font-family: "stixgeneral"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">1</span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 0.802em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.048em; width: 0px;"></span></span></nobr></span>. In other words, the output from the softmax layer can be thought of as a probability distribution.<br />
<br />
<span style="font-family: Georgia, "Times New Roman", serif;">The fact that a softmax layer outputs a probability distribution is rather pleasing. In many problems it's convenient to be able to interpret the output activation </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mi>j</mi><mi>L</mi></msubsup></math>" id="MathJax-Element-213-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: Georgia, "Times New Roman", serif; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2937" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 1.032em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(1.179em 1000.98em 2.699em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.105em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2938" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2939" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2940" style="border: 0px; display: inline; font-family: STIXGeneral; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2941" style="border: 0px; display: inline; font-family: STIXGeneral; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2942" style="border: 0px; display: inline; font-family: STIXGeneral; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.11em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.353em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.497em; width: 0px;"></span></span></nobr></span><span style="font-family: Georgia, "Times New Roman", serif;"> as the network's estimate of the probability that the correct output is </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" id="MathJax-Element-214-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: Georgia, "Times New Roman", serif; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2943" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.297em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.297em;"><span style="border: 0px; clip: rect(1.718em 1000.3em 2.846em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.498em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2944" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2945" style="border: 0px; display: inline; font-family: STIXGeneral; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.002em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.247em; width: 0px;"></span></span></nobr></span><span style="font-family: Georgia, "Times New Roman", serif;">. So, for instance, in the MNIST classification problem, we can interpret </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mi>j</mi><mi>L</mi></msubsup></math>" id="MathJax-Element-215-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: Georgia, "Times New Roman", serif; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2946" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 1.032em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(1.179em 1000.98em 2.699em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.105em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2947" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2948" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.983em;"><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.017em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2949" style="border: 0px; display: inline; font-family: STIXGeneral; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.49em 4.169em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.36em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2950" style="border: 0px; display: inline; font-family: STIXGeneral; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">L<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.434em 1000.3em 4.316em -999.998em); left: 0.493em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.723em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2951" style="border: 0px; display: inline; font-family: STIXGeneral; font-size: 14.4228px; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; display: inline-block; height: 4.022em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.11em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.353em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.497em; width: 0px;"></span></span></nobr></span><span style="font-family: Georgia, "Times New Roman", serif;"> as the network's estimated probability that the correct digit classification is </span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" id="MathJax-Element-216-Frame" role="presentation" style="border: 0px; color: #2a2a2a; direction: ltr; display: inline; float: none; font-family: Georgia, "Times New Roman", serif; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-2952" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.297em;"><span style="border: 0px; display: inline-block; font-size: 20.4px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.297em;"><span style="border: 0px; clip: rect(1.718em 1000.3em 2.846em -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.498em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2953" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2954" style="border: 0px; display: inline; font-family: STIXGeneral; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">j<span style="border: 0px; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; display: inline-block; height: 2.502em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.002em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.247em; width: 0px;"></span></span></nobr></span><span style="font-family: Georgia, "Times New Roman", serif;">.</span></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-10548627840489889902017-12-23T16:33:00.001+05:302017-12-23T16:35:10.731+05:30Install and use IPython Notebook<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="color: blue;">sudo apt-get -y install ipython ipython-notebook</span><br />
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<span style="color: blue;">sudo -H pip install jupyter</span></div>
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Running Jupyter : Execute the following command: </div>
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<span style="color: blue;">jupyter notebook </span></div>
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More details here: <a href="https://www.digitalocean.com/community/tutorials/how-to-set-up-a-jupyter-notebook-to-run-ipython-on-ubuntu-16-04" target="_blank">How To Set Up a Jupyter Notebook to Run IPython on Ubuntu 16.04</a></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-87512257629340702102017-11-10T04:29:00.001+05:302018-09-21T15:10:52.795+05:30Convolutional Neural Networks - CNNs<div dir="ltr" style="text-align: left;" trbidi="on">
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<img border="0" data-original-height="191" data-original-width="720" src="https://www.clarifai.com/static/images/cnn.png" /><a href="https://ujjwalkarn.me/2016/08/11/intuitive-explanation-convnets/" target="_blank">https://ujjwalkarn.me/2016/08/11/intuitive-explanation-convnets/</a></div>
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<b>Convolution: </b>The primary purpose of Convolution in case of a CNNs is to extract features from the input image. Convolution preserves the spatial relationship between pixels by learning image features</div>
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<a href="https://ujwlkarn.files.wordpress.com/2016/07/convolution_schematic.gif?w=268&h=196" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="196" data-original-width="268" src="https://ujwlkarn.files.wordpress.com/2016/07/convolution_schematic.gif?w=268&h=196" /></a></div>
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In the computation above we slide the orange matrix over the original image (green) by 1 pixel (also called ‘stride’) and for every position, we compute element wise multiplication (between the two matrices) and add the multiplication outputs to get the final value which forms a single element of the output matrix (pink). Note that the 3×3 matrix “sees” only a part of the input image in each stride.</div>
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In CNNs terminology, the 3×3 matrix is called a ‘filter‘ or ‘kernel’ or ‘feature detector’ and the matrix formed by sliding the filter over the image and computing the dot product is called the ‘Convolved Feature’ or ‘Activation Map’ or the ‘Feature Map‘. It is important to note that filters acts as feature detectors from the original input image.</div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-28254420626090931762017-11-04T04:36:00.000+05:302017-11-21T21:49:29.420+05:30Machine Learning - Model Evaluation Metrics<div dir="ltr" style="text-align: left;" trbidi="on">
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Confusion Matrix:</h4>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5YsSbkVBUJHt1u586x-lsjZCqe0u5y117ZfysWh02z21coGUQGFEBrqG-LLURA1OQPreoic7B84TH4xZhW7tTkj1fdPepcjr4AJ3HGtG99-urJwcjCLGuCAdv8Rp2lU5pmyAC6w/s1600/Evaluation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="768" data-original-width="1366" height="358" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5YsSbkVBUJHt1u586x-lsjZCqe0u5y117ZfysWh02z21coGUQGFEBrqG-LLURA1OQPreoic7B84TH4xZhW7tTkj1fdPepcjr4AJ3HGtG99-urJwcjCLGuCAdv8Rp2lU5pmyAC6w/s640/Evaluation.png" width="640" /></a></div>
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<b>ROC (Receiver Operating Characteristics) and Area Under Curve (AUC)</b></h4>
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ROC graphs are two-dimensional graphs in which tp rate (true positive rate or recall in above diagram) is plotted on the Y axis and fp rate is plotted on the X axis. An ROC graph depicts relative tradeoffs between benefits (true positives) and costs (false positives). More details: <a href="http://people.inf.elte.hu/kiss/13dwhdm/roc.pdf" target="_blank">An introduction to ROC analysis by Tom Fawcett</a><br />
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For example, when you consider the results of a particular test in two populations, one population with a disease, the other population without the disease, you will rarely observe a perfect separation between the two groups. Indeed, the distribution of the test results will overlap, as shown in the following figure.<br />
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<a href="https://www.medcalc.org/manual/_help/images/roc_intro1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="172" data-original-width="290" src="https://www.medcalc.org/manual/_help/images/roc_intro1.png" /></a></div>
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For every possible cut-off point or criterion value you select to discriminate between the two populations, there will be some cases with the disease correctly classified as positive (TP = True Positive fraction), but some cases with the disease will be classified negative (FN = False Negative fraction). On the other hand, some cases without the disease will be correctly classified as negative (TN = True Negative fraction), but some cases without the disease will be classified as positive (FP = False Positive fraction).</div>
<br />
<ul style="font-family: Roboto, Arial, Helvetica, sans-serif; font-size: 14px; list-style-image: url("/gif/bullet6.png"); margin-bottom: 6px; margin-top: 6px; padding-left: 30px; text-align: justify;">
<li style="margin-left: 20px; margin-top: 6px; padding-left: 0px; text-align: justify;"><i>Sensitivity</i>: probability that a test result will be positive when the disease is present (true positive rate, expressed as a percentage). </li>
<li style="margin-left: 20px; margin-top: 6px; padding-left: 0px; text-align: justify;"><i>Specificity</i>: probability that a test result will be negative when the disease is not present (true negative rate, expressed as a percentage). </li>
<li style="margin-left: 20px; margin-top: 6px; padding-left: 0px; text-align: justify;"><i>Positive likelihood ratio</i>: ratio between the probability of a positive test result given the <i>presence</i> of the disease and the probability of a positive test result given the <i>absence</i> of the disease, i.e. = True positive rate / False positive rate = Sensitivity / (1-Specificity)</li>
<li style="margin-left: 20px; margin-top: 6px; padding-left: 0px; text-align: justify;"><i>Negative likelihood ratio</i>: ratio between the probability of a negative test result given the <i>presence</i> of the disease and the probability of a negative test result given the <i>absence</i> of the disease, i.e. = False negative rate / True negative rate = (1-Sensitivity) / Specificity</li>
<li style="margin-left: 20px; margin-top: 6px; padding-left: 0px; text-align: justify;"><i>Positive predictive value</i>: probability that the disease is present when the test is positive (expressed as a percentage).</li>
<li style="margin-left: 20px; margin-top: 6px; padding-left: 0px; text-align: justify;"><i>Negative predictive value</i>: probability that the disease is not present when the test is negative (expressed as a percentage). </li>
</ul>
<div style="text-align: justify;">
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
In a Receiver Operating Characteristic (ROC) curve the true positive rate (Sensitivity) is plotted in function of the false positive rate (100-Specificity) for different cut-off points. Each point on the ROC curve represents a sensitivity/specificity pair corresponding to a particular decision threshold. A test with perfect discrimination (no overlap in the two distributions) has a ROC curve that passes through the upper left corner (100% sensitivity, 100% specificity). Therefore the closer the ROC curve is to the upper left corner, the higher the overall accuracy of the test - see next diagram.</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://www.medcalc.org/manual/_help/images/roc_intro3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="279" data-original-width="289" src="https://www.medcalc.org/manual/_help/images/roc_intro3.png" /></a></div>
<div>
<br />
The most important metric are the following:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Precisionrecall.svg/500px-Precisionrecall.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="440" height="640" src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Precisionrecall.svg/500px-Precisionrecall.svg.png" width="352" /></a></div>
<br />
<br /></div>
</div>
</div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-29477138315222343612017-11-02T05:59:00.000+05:302018-01-18T23:44:11.082+05:30Deploy a Python app on Google Cloud<div dir="ltr" style="text-align: left;" trbidi="on">
STEPS:<br />
<blockquote class="tr_bq">
TUTORIALDIR=src/[YOUR_PROJECT_ID]/python_gae_quickstart-2017-11-01-23-03<br />
git clone https://github.com/GoogleCloudPlatform/python-docs-samples $TUTORIALDIR<br />
cd $TUTORIALDIR/appengine/standard/hello_world<br />
dev_appserver.py $PWD<br />
gcloud app deploy app.yaml --project [YOUR_PROJECT_ID]</blockquote>
The app runs on: <span style="background-color: white;"><span style="color: purple;">https://[YOUR_PROJECT_ID].appspot.com/</span></span><br />
<br />
<b>Flask App on Google App Engine:</b><br />
<br />
<blockquote class="tr_bq">
cd $TUTORIALDIR/appengine/standard/flask/tutorial<br />
gcloud app deploy app.yaml --project [YOUR_PROJECT_ID]</blockquote>
Run the Flask app on: <span style="color: magenta;">https://[YOUR_PROJECT_ID].appspot.com/form</span><br />
<i><span style="color: #3d85c6;"><span style="color: magenta;">Example: </span>/src/tantal-183814/python_gae_quickstart-2017-11-01-23-03/appengine/standard/flask/tutorial</span></i><br />
<br />
Using the <b>Container Engine</b>:<br />
<br />
<a href="https://cloud.google.com/container-engine/docs/quickstart#optional_hello_app_code_review">https://cloud.google.com/container-engine/docs/quickstart#optional_hello_app_code_review</a><br />
<br />
This example makes use of a web app framework - a<span style="color: #212121; font-family: "roboto" , sans-serif; font-size: 16px;"> web application framework can simplify development by taking care of the details of the interface, letting you focus development effort on your applications features. App Engine includes a simple web application framework called</span><span style="color: #212121; font-family: "roboto" , sans-serif; font-size: 16px;"> </span><a href="http://webapp-improved.appspot.com/" style="box-sizing: inherit; color: #039be5; font-family: Roboto, sans-serif; font-size: 16px; outline: 0px; text-decoration-line: none;">webapp2</a><span style="color: #212121; font-family: "roboto" , sans-serif; font-size: 16px;"> </span><span style="color: #212121; font-family: "roboto" , sans-serif; font-size: 16px;">- a lightweight framework that allows you quickly build simple web applications for the Python 2.7 runtime.</span><br />
<div style="box-sizing: inherit; color: #212121; font-family: Roboto, sans-serif; font-size: 16px; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
webapp2 is compatible with the WSGI standard for Python web applications. You don't have to use webapp2 to write Python applications for App Engine. Other web application frameworks, such as <a href="http://www.djangoproject.com/" style="box-sizing: inherit; color: #039be5; outline: 0px; text-decoration-line: none;">Django</a>, work with App Engine, and App Engine supports any Python code that uses the CGI standard. The webapp2 project, by Rodrigo Moraes, started as a fork of the App Engine <a href="https://cloud.google.com/appengine/docs/standard/python/tools/webapp" style="box-sizing: inherit; color: #039be5; outline: 0px; text-decoration-line: none;">webapp</a> framework, which was used by the Python 2.5 runtime. webapp2 includes a number of features that make developing web applications easier, such as improved support for URI routing, session management and localization. The Python 2.7 runtime uses webapp2, and the project is maintained externally to App Engine. It is supported, but not maintained, by Google.</div>
<div style="box-sizing: inherit; margin-bottom: 16px; margin-top: 16px; padding: 0px;">
<div style="color: #212121; font-family: roboto, sans-serif; font-size: 16px;">
For more information about webapp2, see the <a href="http://webapp-improved.appspot.com/" style="box-sizing: inherit; color: #039be5; outline: 0px; text-decoration-line: none;">official documentation</a>.</div>
<div style="color: #212121; font-family: roboto, sans-serif; font-size: 16px;">
<b><br /></b></div>
<div style="color: #212121; font-family: roboto, sans-serif; font-size: 16px;">
<b>RUNNING Django + setup a MySQL database</b> instance on AppEngine: </div>
<span style="color: #212121; font-family: "roboto" , sans-serif;"><a href="https://cloud.google.com/python/django/appengine#configure_the_database_settings">https://cloud.google.com/python/django/appengine#configure_the_database_settings</a></span><br />
<span style="color: #212121; font-family: "roboto" , sans-serif;"><br /></span></div>
</div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-53445345034742271652017-11-01T16:16:00.001+05:302017-11-01T16:17:02.560+05:30Google Cloud - Natural Language API<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
Apart from the Quickstart here:<br />
<a href="https://cloud.google.com/natural-language/docs/">https://cloud.google.com/natural-language/docs/</a><br />
<br />
I used this code:<br />
<a href="https://raw.githubusercontent.com/GoogleCloudPlatform/python-docs-samples/master/language/cloud-client/v1/quickstart.py">https://raw.githubusercontent.com/GoogleCloudPlatform/python-docs-samples/master/language/cloud-client/v1/quickstart.py</a><br />
<br />
I had to do a few additional steps including:<br />
<br />
Setting up Google Application Default Credentials - including setting the environment variable GOOGLE_APPLICATION_CREDENTIALS<br />
<br />
Well described here:<br />
<a href="https://developers.google.com/identity/protocols/application-default-credentials">https://developers.google.com/identity/protocols/application-default-credentials</a><br />
<br />
Then I had to disable unwanted warnings based on the details here:<br />
<a href="https://urllib3.readthedocs.io/en/latest/advanced-usage.html#ssl-warnings">https://urllib3.readthedocs.io/en/latest/advanced-usage.html#ssl-warnings</a><br />
<br />
<br /></div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-20920406759202680232017-10-23T15:43:00.000+05:302017-10-23T17:03:37.335+05:30Google BigQuery & Apache Hive<div dir="ltr" style="text-align: left;" trbidi="on">
<h4 style="text-align: left;">
Google BIGQUERY <span style="font-weight: normal;">is a fast, economical and fully-managed enterprise data warehouse for large-scale data analytics. </span><span style="font-weight: normal;">Details of querying your custom table in BigQuery:</span></h4>
<a href="https://cloud.google.com/bigquery/quickstart-web-ui">https://cloud.google.com/bigquery/quickstart-web-ui</a><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://mscoco.org/static/images/sponsors-GCP.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://mscoco.org/static/images/sponsors-GCP.png" data-original-height="261" data-original-width="800" height="104" width="320" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
The <b>Apache Hive</b>™ data warehouse software facilitates reading, writing, and managing large datasets residing in distributed storage and queried using SQL syntax. Built on top of Apache Hadoop™, Hive provides the following features:<br />
<ul style="text-align: left;">
<li>Tools to enable easy access to data via SQL, thus enabling data warehousing tasks such as extract/transform/load (ETL), reporting, and data analysis.</li>
</ul>
<ul style="text-align: left;">
<li>A mechanism to impose structure on a variety of data formats</li>
</ul>
<ul style="text-align: left;">
<li>Access to files stored either directly in Apache HDFS™ or in other data storage systems such as Apache HBase™</li>
</ul>
<ul style="text-align: left;">
<li>Query execution via Apache Tez™, Apache Spark™, or MapReduce</li>
</ul>
<ul style="text-align: left;">
<li>Procedural language with HPL-SQL</li>
</ul>
<ul style="text-align: left;">
<li>Sub-second query retrieval via Hive LLAP, Apache YARN and Apache Slider.</li>
</ul>
<br />
More details on getting started: <a href="https://cwiki.apache.org/confluence/display/Hive/GettingStarted">https://cwiki.apache.org/confluence/display/Hive/GettingStarted</a><br />
<br /></div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-34789827503958112142017-10-22T03:24:00.001+05:302017-10-23T04:06:12.371+05:30Spark SQL & 3rd Party Spark Machine Learning libraries<div dir="ltr" style="text-align: left;" trbidi="on">
<h1 style="border-bottom: 1px solid rgb(234, 236, 239); box-sizing: border-box; color: #24292e; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; line-height: 1.25; margin-bottom: 16px; margin-left: 0px; margin-right: 0px; margin-top: 0px !important; padding-bottom: 0.3em;">
Spark machine learning inventory</h1>
<a href="https://github.com/claesenm/spark-ml-inventory">https://github.com/claesenm/spark-ml-inventory</a><br />
<h1 style="box-sizing: border-box; color: #159957; font-family: "Open Sans", "Helvetica Neue", Helvetica, Arial, sans-serif; font-weight: normal; margin: 0px 0px 1rem;">
</h1>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://spark.apache.org/docs/0.9.1/img/spark-logo-hd.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="163" data-original-width="254" src="https://spark.apache.org/docs/0.9.1/img/spark-logo-hd.png" /></a></div>
<br />
<ul style="text-align: left;">
<li><span style="background-color: white; color: #24292e; font-family: , "blinkmacsystemfont" , "segoe ui" , "helvetica" , "arial" , sans-serif , "apple color emoji" , "segoe ui emoji" , "segoe ui symbol"; font-size: 16px;">Apache Spark & Python (pySpark) tutorials for Big Data Analysis and Machine Learning as IPython / Jupyter notebooks: </span><span style="color: #24292e; font-family: , "blinkmacsystemfont" , "segoe ui" , "helvetica" , "arial" , sans-serif , "apple color emoji" , "segoe ui emoji" , "segoe ui symbol";"><a href="http://jadianes.me/spark-py-notebooks/">http://jadianes.me/spark-py-notebooks/</a></span></li>
<li>MLlib: Classification with Logistic Regression: <a href="https://github.com/jadianes/spark-py-notebooks/blob/master/nb8-mllib-logit/nb8-mllib-logit.ipynb">https://github.com/jadianes/spark-py-notebooks/blob/master/nb8-mllib-logit/nb8-mllib-logit.ipynb</a></li>
</ul>
<div>
<br /></div>
<h3 style="text-align: left;">
Spark SQL and Dataframes: <a href="https://www.codementor.io/jadianes/python-spark-sql-dataframes-du107w74i" target="_blank">Python and Spark</a></h3>
<ul style="text-align: left;">
</ul>
Steps:<br />
<br />
Getting the Data and Creating the RDD<br />
<br />
<span style="background-color: white; color: #76a5af;">import urllib</span><br />
<span style="background-color: white; color: #76a5af;">f = urllib.urlretrieve ("http://kdd.ics.uci.edu/databases/kddcup99/kddcup.data_10_percent.gz", "kddcup.data_10_percent.gz")</span><br />
<span style="background-color: white; color: #76a5af;">data_file = "./kddcup.data_10_percent.gz"</span><br />
<span style="background-color: white; color: #76a5af;">raw_data = sc.textFile(data_file).cache()</span><br />
<br />
<br />
Step 2: Create a Dataframe<br />
<br />
<b>A Spark DataFrame is a distributed collection of data organized into named columns. It is conceptually equivalent to a table in a relational database </b>or a data frame in R or Pandas. They can be constructed from a wide array of sources such as an existing RDD in our case.<br />
<br />
The entry point into all SQL functionality in Spark is the SQLContext class. To create a basic instance, all we need is a SparkContext reference.<br />
<br />
from pyspark.sql import SQLContext<br />
sqlContext = SQLContext(sc)<br />
<br />
<b>Inferring the Schema: </b>With a SQLContext, we are ready to create a DataFrame from our existing RDD. But first we need to tell Spark SQL the schema in our data.<br />
<br />
Step 3:<br />
<br />
<div style="box-sizing: inherit; color: #444444; font-family: "Open Sans", sans-serif; font-size: 18px; letter-spacing: 0.1px; line-height: 32px; margin-bottom: 26px;">
Once we have our RDD of <code style="background-color: #f8f8f8; border-radius: 4px; box-sizing: inherit; color: #c7254e; font-family: monospace, monospace; font-size: 14px; padding: 4px 6px; word-break: normal;">Row</code> we can infer and register the schema.</div>
<pre style="background-color: #f8f8f8; border-radius: 4px; border: 1px solid rgba(222, 229, 231, 0.6); box-sizing: inherit; display: inline-block; font-family: monospace, monospace; font-size: 14px; letter-spacing: 0.1px; margin-bottom: 31.2px; margin-left: -21.5938px; overflow-x: scroll; padding: 1.4em; width: 763.188px;"><code class="language-python hljs" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-radius: 4px; box-sizing: inherit; color: inherit; display: block; font-family: monospace, monospace; line-height: 22px; overflow-x: auto; padding: 0px; word-break: normal;">interactions_df = sqlContext.createDataFrame(row_data)
interactions_df.registerTempTable(<span class="hljs-string" style="box-sizing: inherit; color: #dd1144;">"interactions"</span>)
</code></pre>
<div style="box-sizing: inherit; color: #444444; font-family: "Open Sans", sans-serif; font-size: 18px; letter-spacing: 0.1px; line-height: 32px; margin-bottom: 26px;">
Now we can run SQL queries over our data frame that has been registered as a table.</div>
<div style="box-sizing: inherit; color: #444444; font-family: "Open Sans", sans-serif; font-size: 18px; letter-spacing: 0.1px; line-height: 32px; margin-bottom: 26px;">
<code class="language-python hljs" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-radius: 4px; box-sizing: inherit; color: inherit; display: block; font-family: monospace, monospace; font-size: 14px; letter-spacing: 0.1px; line-height: 22px; overflow-x: auto; padding: 0px; white-space: pre; word-break: normal;"><span class="hljs-comment" style="box-sizing: inherit; color: #999988; font-style: italic;"><br class="Apple-interchange-newline" /># Select tcp network interactions with more than 1 second duration and no transfer from destination</span>
tcp_interactions = sqlContext.sql(<span class="hljs-string" style="box-sizing: inherit; color: #dd1144;">"""
SELECT duration, dst_bytes FROM interactions WHERE protocol_type = 'tcp' AND duration > 1000 AND dst_bytes = 0
"""</span>)
tcp_interactions.show()</code></div>
<div>
<div style="box-sizing: inherit; color: #444444; font-family: "Open Sans", sans-serif; font-size: 18px; letter-spacing: 0.1px; line-height: 32px; margin-bottom: 26px;">
The results of SQL queries are RDDs and support all the normal RDD operations.</div>
<pre style="background-color: #f8f8f8; border-radius: 4px; border: 1px solid rgba(222, 229, 231, 0.6); box-sizing: inherit; display: inline-block; font-family: monospace, monospace; font-size: 14px; letter-spacing: 0.1px; margin-bottom: 31.2px; margin-left: -21.5938px; overflow-x: scroll; padding: 1.4em; width: 763.188px;"><code class="language-python hljs" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-radius: 4px; box-sizing: inherit; color: inherit; display: block; font-family: monospace, monospace; line-height: 22px; overflow-x: auto; padding: 0px; word-break: normal;"><span class="hljs-comment" style="box-sizing: inherit; color: #999988; font-style: italic;"># Output duration together with dst_bytes</span>
tcp_interactions_out = tcp_interactions.rdd.map(<span class="hljs-keyword" style="box-sizing: inherit; color: #333333; font-weight: bold;">lambda</span> p: <span class="hljs-string" style="box-sizing: inherit; color: #dd1144;">"Duration: {}, Dest. bytes: {}"</span>.format(p.duration, p.dst_bytes))
<span class="hljs-keyword" style="box-sizing: inherit; color: #333333; font-weight: bold;">for</span> ti_out <span class="hljs-keyword" style="box-sizing: inherit; color: #333333; font-weight: bold;">in</span> tcp_interactions_out.collect():
<span class="hljs-keyword" style="box-sizing: inherit; color: #333333; font-weight: bold;">print</span> ti_out</code></pre>
<br />
<br />
One thing to remember: You can’t map a dataframe, but you can convert the dataframe to an RDD and map that by doing spark_df.rdd.map(). Prior to Spark 2.0, spark_df.map would alias to spark_df.rdd.map(). With Spark 2.0, you must explicitly call .rdd first.</div>
<br />
<br />
<div style="box-sizing: inherit; color: #444444; font-family: "Open Sans", sans-serif; font-size: 18px; letter-spacing: 0.1px; line-height: 32px; margin-bottom: 26px;">
<br /></div>
<div style="box-sizing: inherit; color: #444444; font-family: "Open Sans", sans-serif; font-size: 18px; letter-spacing: 0.1px; line-height: 32px; margin-bottom: 26px;">
<br /></div>
</div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-81902567186817745932017-10-21T05:16:00.000+05:302017-10-24T04:39:55.689+05:30HDFS (Hadoop), Scikit-Learn & Apache Spark MLlib <div dir="ltr" style="text-align: left;" trbidi="on">
On Linux: Ubuntu 14.04.5 LTS, Release: 14.04, trusty.<br />
<div>
<br /></div>
Apache Hadoop is an open source software framework that enables large data sets to be broken up into blocks, distributed to multiple servers for storage and processing. Hadoop’s strength comes from a server network – known as a Hadoop cluster – that can process data much more quickly than a single machine. The non-profit Apache Software Foundation supports the free open source Hadoop project, but commercial versions have become very common.<br />
<br />
The Hadoop Distributed File System (HDFS) is the place in a Hadoop cluster that you store data. Built for data-intensive applications, the HDFS is designed to run on clusters of inexpensive commodity servers. HDFS is optimized for high performance, read intensive operations and resilient to failures in the cluster. It does not prevent failures but is unlikely to lose data, since by default HDFS makes multiple copies of each of its data blocks.<br />
<br />
<span style="color: orange;"><span style="color: orange; font-weight: bold;">Hadoop</span></span> does <b><span style="color: orange;">batch processing</span></b> i.e processing of blocks of data already stored over a period of time. Initially Hadoop's MapReduce technique was the best framework for processing data in batches. Spark is an open-source <b>cluster computing framework</b> for real-time processing. <b><span style="color: red;">Spark</span>'s</b> additional functionality is that it can <span style="color: red;"><b>process data in real time</b></span> and since it was built on top of Hadoop MapReduce and it extends the MapReduce model to efficiently use more types of computations it is also about 100 times faster than Hadoop MapReduce in batch processing large data sets.<br />
<br />
Spark can create distributed datasets from any file stored in the Hadoop distributed filesystem (HDFS) or other storage systems supported by the Hadoop APIs (including your local filesystem, Amazon S3, Cassandra, Hive, HBase, etc.). Spark does not require Hadoop; it simply has support for storage systems implementing the Hadoop APIs. Spark supports text files, SequenceFiles etc and any other Hadoop InputFormat.<br />
<br />
More differences and Spark details are here: <a href="https://www.edureka.co/blog/spark-tutorial/">https://www.edureka.co/blog/spark-tutorial/</a><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNTTXsAF58NN9uH-RW2xLDlZaFbgxDkm9TfMFDJPwDC2gqWTRMNcc8co7Cdsoedb4YECRtcJoPhf-V8aPtFrt8Xkgahp2Gx63CVq4ie2F-vFonDYAy22NUDrSoAk-_x43frm8TiA/s1600/Spark.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="430" data-original-width="582" height="295" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNTTXsAF58NN9uH-RW2xLDlZaFbgxDkm9TfMFDJPwDC2gqWTRMNcc8co7Cdsoedb4YECRtcJoPhf-V8aPtFrt8Xkgahp2Gx63CVq4ie2F-vFonDYAy22NUDrSoAk-_x43frm8TiA/s400/Spark.png" width="400" /></a></div>
<br />
<br />
<a href="https://www.digitalocean.com/community/tutorials/how-to-install-hadoop-in-stand-alone-mode-on-ubuntu-16-04" target="_blank"><b>Install Hadoop in Stand-Alone Mode on Ubuntu 16.04</b></a><br />
<br />
Once installed run it as:<br />
<span style="background-color: rgba(0 , 0 , 0 , 0.05); color: #3a3a3a; font-family: monospace; font-size: 14px; white-space: pre;">/usr/local/hadoop/bin/hadoop</span><br />
<span style="background-color: rgba(0 , 0 , 0 , 0.05); color: #3a3a3a; font-family: monospace; font-size: 14px; white-space: pre;"><br /></span>
<br />
Scikit-Learn ML Examples:<br />
<a href="http://scikit-learn.org/stable/auto_examples/index.html#">http://scikit-learn.org/stable/auto_examples/index.html#</a><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZWIL6OaTLsvMeyliwAbLE1GhdtCKC0pS3BEGJ9Tg9BvMFcEozC2QCR1b2eLkf_jI2Dk06CsvKB__Gh20cY7q7854x7K326Ww7dmmq6H3HiQTBVE16_8YIAn1UWKcmmFKnEREC2g/s1600/Screenshot+from+2017-10-21+13%253A51%253A17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="768" data-original-width="1366" height="358" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZWIL6OaTLsvMeyliwAbLE1GhdtCKC0pS3BEGJ9Tg9BvMFcEozC2QCR1b2eLkf_jI2Dk06CsvKB__Gh20cY7q7854x7K326Ww7dmmq6H3HiQTBVE16_8YIAn1UWKcmmFKnEREC2g/s640/Screenshot+from+2017-10-21+13%253A51%253A17.png" width="640" /></a></div>
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<br />
<br />
<a name='more'></a><br />
<br />
<h4 style="text-align: left;">
Spark Examples:</h4>
<a href="https://spark.apache.org/examples.html">https://spark.apache.org/examples.html</a><br />
<br />
<pre class="pre codeblock hljs" style="background: rgb(240, 240, 240); border: 1px dotted rgb(102, 102, 102); box-sizing: border-box; color: #444444; font-family: monospace, serif; font-size: 13px; margin-bottom: 10px; margin-top: 10px; overflow: auto; padding: 5px; word-wrap: normal;">Welcome to
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/ __/__ ___ _____/ /__
_\ \/ _ \/ _ `/ __/ '_/
/__ / .__/\_,_/_/ /_/\_\ version ...
/_/</pre>
<div>
<br /></div>
There was a problem such as the following while running pyspark<br />
<blockquote class="tr_bq">
</blockquote>
<blockquote class="tr_bq">
Exception in thread "main" java.lang.UnsupportedClassVersionError: org/apache/spark/launcher/Main : Unsupported major.minor version 52.0</blockquote>
<br />
Apache Maven and JDK 8 had to be installed. Details here:<br />
<a href="https://www.digitalocean.com/community/tutorials/how-to-install-java-with-apt-get-on-ubuntu-16-04">https://www.digitalocean.com/community/tutorials/how-to-install-java-with-apt-get-on-ubuntu-16-04</a><br />
<br />
Also another problem to keep in mind from some of the Spark MLib code on the website to add the context and session variables:<br />
<br />
<blockquote class="tr_bq">
from pyspark.context import SparkContext<br />
from pyspark.sql.session import SparkSession<br />
sc = SparkContext('local')<br />
spark = SparkSession(sc)</blockquote>
<br />
<br />
<h3 style="text-align: left;">
<b><span style="font-size: large;"><span style="color: red;">Big Data with Apache Spark</span></span></b></h3>
<br />
<ul style="text-align: left;">
<li><b><a href="https://github.com/dipanjanS/BerkeleyX-CS100.1x-Big-Data-with-Apache-Spark/blob/master/Week%205%20-%20Introduction%20to%20Machine%20Learning%20with%20Apache%20Spark/lab4_machine_learning_student.ipynb" target="_blank"><span style="font-size: large;">Machine Learning using Spark</span></a></b></li>
<li><b><span style="font-size: large;"><a href="https://github.com/dipanjanS/BerkeleyX-CS100.1x-Big-Data-with-Apache-Spark/blob/master/Week%202%20-%20Introduction%20to%20Apache%20Spark/lab1_word_count_student.ipynb" target="_blank">Python Code to create RDD, Map and Collect etc.</a></span></b></li>
</ul>
<br />
<br />
HDFS with Spark: <a href="https://cbw.sh/spark.html">https://cbw.sh/spark.html</a><br />
<br />
Setting Up Your Environment - In order to use HDFS and Spark, you first need to configure your environment so that you have access to the required tools. The easiest way to do this is to modify the <b>.bashrc </b>configuration file in your home directory.</div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-56794363192949041372017-07-08T18:17:00.002+05:302017-07-12T15:07:53.014+05:30Multi-Class Classification with the Keras Deep Learning / TensorFlow<div dir="ltr" style="text-align: left;" trbidi="on">
Ubuntu 14.04<br />
<br />
TensorFlow: <a href="https://www.tensorflow.org/">https://www.tensorflow.org</a><br />
<br />
I had to install Numpy, Pandas: <a class="" href="http://pandas.pydata.org/">http://pandas.pydata.org/</a> , Ski-Kit: <a href="http://scikit-learn.org/">http://scikit-learn.org/</a><br />
<br />
pip install -U scikit-learn<br />
<br />
Complete methodology utilized is available on Jason Brownlee's blog here:<br />
<a href="http://machinelearningmastery.com/multi-class-classification-tutorial-keras-deep-learning-library/">http://machinelearningmastery.com/multi-class-classification-tutorial-keras-deep-learning-library/</a><br />
<br />
<br /></div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-64293588279619226882016-01-19T12:26:00.002+05:302017-11-13T19:58:46.275+05:30Simple C++ Makefile on Linux<div dir="ltr" style="text-align: left;" trbidi="on">
Seems a pain to get started with makefiles for C++ code on Ubuntu.<br />
<br />
First create and name the file 'Makefile' in the same folder as your .h and .cpp files<br />
<br />
Second copy in the following contents and change the .cpp filenames:<br />
<br />
<span style="color: #0b5394;">CC=g++</span><br />
<span style="color: #0b5394;">CFLAGS=-c -Wall</span><br />
<span style="color: #0b5394;">LDFLAGS=</span><br />
<span style="color: #0b5394;">SOURCES=Test.cpp Foo.cpp Bar.cpp</span><br />
<span style="color: #0b5394;">OBJECTS=$(SOURCES:.cpp=.o)</span><br />
<span style="color: #0b5394;">EXECUTABLE=Test</span><br />
<span style="color: #0b5394;"><br /></span> <span style="color: #0b5394;">all: $(SOURCES) $(EXECUTABLE)</span><br />
<span style="color: #0b5394;"> </span><br />
<span style="color: #0b5394;">$(EXECUTABLE): $(OBJECTS)</span><br />
<span style="color: #cc0000;"><span style="color: #cc0000; white-space: pre;">Tab </span></span><span style="color: #0b5394;">$(CC) $(LDFLAGS) $(OBJECTS) -o $@</span><br />
<span style="color: #0b5394;"><br /></span> <span style="color: #0b5394;">.cpp.o:</span><br />
<span style="color: #cc0000; white-space: pre;"><tab>Tab </tab></span><span style="color: #0b5394;">$(CC) $(CFLAGS) $< -o $@</span><br />
<br />
<br />
* Standard format is: g++ test1.cpp -o test1</div>
shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-72056623730715428442015-10-27T23:07:00.002+05:302015-10-28T03:57:58.447+05:30IKEA Response - Glass Tumbler Explodes.<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">I got the following response from IKEA after the incident: </span></div>
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<a href="http://shamitb.blogspot.se/2015/10/ikea-glass-tumbler-explodes.html" style="color: #888888; font-family: 'Trebuchet MS', Trebuchet, Verdana, sans-serif; font-size: 22px; text-decoration: none;">IKEA Glass Tumbler Explodes.</a></div>
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<span style="color: #cc0000; font-family: Times New Roman, serif;"><i><span style="font-size: 12pt;">However my response would be </span>I underwent this agony after IKEA selling me this untested/defective good (explosive?) and this untoward accident which happened at our premises and might have caused so much more damage. There was hardly any thermal shock - after almost 5-10 mins after pouring water from the tap (cold) the glass exploded with a loud sound and scattered around.</i></span><br />
<span style="color: #cc0000;"><span style="font-family: Times New Roman, serif;"><i><br /></i></span><span style="font-family: Times New Roman, serif;"><i>I bought the product thinking it is robust and not so fragile that merely trying to drink water it will explode into pieces. Also IKEA needs to ensure this product is checked/evaluated for safety standards or else any customer could get hurt - knowingly selling this product would amount to <b>willful negligence</b>. </i></span></span><br />
<span style="color: #cc0000;"><span style="font-family: Times New Roman, serif;"><i><br /></i></span>
<span style="font-family: Times New Roman, serif;"><i>As a friend of mine states consider an accident happens with my car due to a fault in the system and I am exposed to danger. Can the car company just educate me and their job is done? </i></span></span><br />
<span style="color: #cc0000;"><span style="font-family: Times New Roman, serif;"><i><br /></i></span>
<span style="font-family: Times New Roman, serif;"><i>CE mark, ISI mark etc ensure the quality standard. IKEA is trying to tell it can not be checked better, by that logic in very cold country and very hot country where inside air and outside air there is big temperature gap windows must break, making NO windows (car wind screens) good enough to withstand temp gradient, that is purely unacceptable.</i></span></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Hello Shamit!<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Thank you for your e-mail.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">We understand you are worried and concerned over safety risks when using our glass.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Though we need to explain how they are manufactured and what type of glass are being sold on the market.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Here follows a short explanation of the difference between ordinary, non-tempered and tempered glass.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">The ordinary, non-tempered glass have thermal shock resistance, sudden temperature change, up to 80° Celsius.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Difference between room temperature of 20° Celsius and boiling water, 100° Celsius.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">If broken, it can break in big, sharp pieces that can cause cut and burn injuries. See photo below.<u></u><u></u></span></div>
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<span style="color: #5b9bd5; font-family: Verdana, sans-serif; font-size: 10pt;"><img alt="cid:image001.jpg@01D110DE.3E8F7C30" class="CToWUd a6T" height="315" src="https://mail.google.com/mail/u/0/?ui=2&ik=d3e78a80fd&view=fimg&th=150aa351114221d1&attid=0.1&disp=emb&attbid=ANGjdJ_vd1ndWJoa5E4gQP1jTxJmelKHlA5JR_QgVwCgmZdiEKuw-A3gnjrpPgIr1MPOq8fWwRzUpYfS9xhCFtVk8m3QZzDNdzKBlORIonL3BjlaBdO0l8Mjwv4_mWg&sz=w840-h630&ats=1445964622293&rm=150aa351114221d1&zw&atsh=1" style="cursor: pointer; outline: 0px;" tabindex="0" width="420" /></span><span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;"><u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">(Breaking pattern of ordinary glass, not tempered)<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">BUT - tempered glass has higher thermal shock resistance, above 120° Celsius and <b>if broken, it breaks in small square pieces that are not sharp and<u></u><u></u></b></span></div>
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<b><span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">do not pose any risk of cut burn or burn injury.</span></b><span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;"><u></u><u></u></span></div>
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<span style="color: #5b9bd5;"><img alt="Test result 180° C.JPG" class="CToWUd a6T" height="289" src="https://mail.google.com/mail/u/0/?ui=2&ik=d3e78a80fd&view=fimg&th=150aa351114221d1&attid=0.2&disp=emb&attbid=ANGjdJ9dg9XdB8mtI3TnMKd9COVgvqpx_9TvpXAIO19Q02YdzG9hflQBiK2TtHVJ3GihG52NsIvwsK-yWMef0z6eHPLWgilZNIT29O4n52BJax319t1kQUHUCflEY4s&sz=w844-h578&ats=1445964622294&rm=150aa351114221d1&zw&atsh=1" style="cursor: pointer; outline: 0px;" tabindex="0" width="422" /></span><span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;"><u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">(Breaking pattern of tempered glass)<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">However, all glass is fragile, including tempered and if exposed to mechanical damage, knocks, scratched or impact, glass can break spontaneously days or weeks after being exposed.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">That is why all glass is to be handled with caution and avoid damage at all time. <i>This is not a unique attribute only for IKEA products, but the same no matter which manufacturer.</i><u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">IKEA have choose to use tempered glass in POKAL to minimize the risk of getting injured if and when POKAL breaks.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">We hope this brings some answers to your questions, and you now understand more how glass products works.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">If you or your wife sustained injuries and were in need of medical attention, please let us know. You then need to contact your insurance company and have the injuries documented.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">You are also welcome to send in the remainings of the glass for investigation to our technicians.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">If so, please respond with your full contact details and we are able to help you further.<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Kindest regards,<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">Mattias<u></u><u></u></span></div>
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<span style="color: #1f4e79; font-family: Verdana, sans-serif; font-size: 10pt;">IKEA customer service</span></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-69563080203886896432015-10-27T05:22:00.002+05:302015-10-28T05:03:33.907+05:30IKEA Glass Tumbler Explodes.<div dir="ltr" style="text-align: left;" trbidi="on">
<b><span style="font-size: large;">An IKEA glass tumbler (we bought in </span></b><b><span style="font-size: large;">Gothenburg, </span></b><b><span style="font-size: large;">Sweden) exploded on our dining table tonight.</span></b><br />
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<span style="font-size: large; margin-left: 1em; margin-right: 1em; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZmMkSDYrXwrrmRM4GuO24Ze-Gxy_UW5_gtpUlfpS4j71mzCQVgwch1KHA3P-94yyo6o3OBhHNUiFPMtT1kTh90w1RyfsKMj8nuaQfK7ZkXceY4pK9z-fTkyLEH16UN8VRzMDY9w/s1600/DSCN1069.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZmMkSDYrXwrrmRM4GuO24Ze-Gxy_UW5_gtpUlfpS4j71mzCQVgwch1KHA3P-94yyo6o3OBhHNUiFPMtT1kTh90w1RyfsKMj8nuaQfK7ZkXceY4pK9z-fTkyLEH16UN8VRzMDY9w/s640/DSCN1069.JPG" width="640" /></a></span><br />
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<span style="font-size: large;">We had a few guests at the table and they were having dinner when the glass exploded and shattered into small pieces and minute fragments. It is sheer luck that nobody was hurt.</span><br />
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<span style="font-size: large;">Cold tap water was poured into the IKEA glass tumbler (which was lying unused inside the cupboard) and was offered to the guests. This happened within 5 minutes after pouring the cold tap water. With a loud thud sound (as if it had been shot), the glass exploded and shattered. People could have been badly hurt because of what happened and such products must not be produced (or procured from sub-standard sources) and sold without proper quality testing.</span><br />
<span style="font-size: large;"><br /></span><span style="font-size: large;">This was so totally ridiculous, that buying glass products from IKEA will be a risky proposition hereon. I am apprehensive that guests could have been severely hurt and it makes me shudder (what if shards or </span><span style="font-size: large;">particles</span><span style="font-size: large;"> of </span><span style="font-size: large;">glass had entered the eyes etc).</span><br />
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<b><span style="font-size: large;">The glass is of the same make as in the image below (Product: <a href="http://www.ikea.com/gb/en/catalog/products/10270478/" target="_blank">POKAL Glass</a>) and was bought at IKEA Gothenburg (IKEA Bäckebol):</span></b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHb2PKtRny-CH_2IuqetUCJTM0Qg2gIbT1akHQ4DhlAeC3p8kYo5NMZfe1EirVDWaoNIynhRR4YKPAsTIKlWmaJvL7-ZV2lP5d_Q43xLYVVilditP8SQx60e8d7gSgVz4PQE5IXA/s1600/DSCN1071.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHb2PKtRny-CH_2IuqetUCJTM0Qg2gIbT1akHQ4DhlAeC3p8kYo5NMZfe1EirVDWaoNIynhRR4YKPAsTIKlWmaJvL7-ZV2lP5d_Q43xLYVVilditP8SQx60e8d7gSgVz4PQE5IXA/s640/DSCN1071.JPG" width="640" /></span></a></div>
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<b><span style="font-size: large;">Images below - After the glass exploded and the pieces shattered and scattered everywhere on our dinner table:</span></b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoRSp48Rz569uvL1NSzduVNC414pQS0jAY-FP-soy_iTBG2pC4FXBPF9OMBmfz3vuWJJTEXnuaaW0lBKxdapwGJvzSsfqrGyJZ6IkXo8PnqZgnDBVBRERnEnoiIRglMiDBMhG75Q/s1600/DSCN1067.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoRSp48Rz569uvL1NSzduVNC414pQS0jAY-FP-soy_iTBG2pC4FXBPF9OMBmfz3vuWJJTEXnuaaW0lBKxdapwGJvzSsfqrGyJZ6IkXo8PnqZgnDBVBRERnEnoiIRglMiDBMhG75Q/s640/DSCN1067.JPG" width="640" /></span></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZmMkSDYrXwrrmRM4GuO24Ze-Gxy_UW5_gtpUlfpS4j71mzCQVgwch1KHA3P-94yyo6o3OBhHNUiFPMtT1kTh90w1RyfsKMj8nuaQfK7ZkXceY4pK9z-fTkyLEH16UN8VRzMDY9w/s1600/DSCN1069.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZmMkSDYrXwrrmRM4GuO24Ze-Gxy_UW5_gtpUlfpS4j71mzCQVgwch1KHA3P-94yyo6o3OBhHNUiFPMtT1kTh90w1RyfsKMj8nuaQfK7ZkXceY4pK9z-fTkyLEH16UN8VRzMDY9w/s640/DSCN1069.JPG" width="640" /></span></a></div>
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<b><span style="font-size: large;">Images above - After the glass exploded and the pieces shattered everywhere on our dinner table.</span></b><br />
<b><span style="font-size: large;"><br /></span></b><b><span style="color: #cc0000; font-size: large;">Update - IKEA has taken cognizance:</span></b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIuzpAdVPW76Oo27BzC7V_zqt0Z7XmRxT1H-bSpmvie5iQIIDsIDMQsKmtshZScu43FXN-3Clv5zk1AOzQA3RQ5Gu5fSxUjPgBMjXfM9pFyjoECbrCkc22n44RhDb_5PIFMwrMkg/s1600/Fullscreen+capture+27-10-2015+010718+PM.bmp.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="331" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIuzpAdVPW76Oo27BzC7V_zqt0Z7XmRxT1H-bSpmvie5iQIIDsIDMQsKmtshZScu43FXN-3Clv5zk1AOzQA3RQ5Gu5fSxUjPgBMjXfM9pFyjoECbrCkc22n44RhDb_5PIFMwrMkg/s640/Fullscreen+capture+27-10-2015+010718+PM.bmp.jpg" width="640" /></a></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-38166697867469177462015-10-11T13:51:00.000+05:302015-10-11T13:51:04.900+05:30Writing a popular science article.<div dir="ltr" style="text-align: left;" trbidi="on">
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-46888066606162953782015-09-12T19:01:00.000+05:302015-09-15T02:59:30.918+05:30Quantum Contrapt - Part 01<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Trebuchet MS, sans-serif; font-size: x-small;">First short story in the 'CSlueth files' series. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: x-small;">Note: This is purely a work of fiction!</span></div>
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<b style="font-family: 'Trebuchet MS', sans-serif;">1. The dead body </b></div>
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<i style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">Wednesday, Chalmers tekniska hogskola, 6:30 PM </i><br />
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<span style="font-family: 'Trebuchet MS', sans-serif;">He had just walked out of the classroom in the V building. For the past forty five minutes he had been coding an exciting dynamic systems simulation after having completed a late and rather boring simulation exercises session class. He had to take the road in front of the Chalmers Bibliotek library as he wanted to meet a friend staying nearby. As he walked past the Information Sciences department building there was a gust of cold winter wind that struck his face and he raised the collars of this jacket, sinking in his chin into its warm, furry interiors. It was December and it would get dark really early. It was dark now and hardly anyone else around. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">He hummed a tune as he ran up the flight of stairs, the library was at an elevated level from the V building. He found himself humming some recent tune he had heard on the YouTube channel, a catchy tune for an aerated drink or something. As he cleared the last step and reached the flat expanse beside the Kemi huset he heard someone moaning almost inaudibly and then the moaning stopped. Tony felt a chill go down his spine as he spotted a motionless dark mass about a feet or two away on the ground. As he walked gingerly ahead he saw that it was the body of a person, seemingly middle-aged and slightly heavily built, now slumped on the ground. His right hand was clutching a piece of rock and was now jutting outward almost perpendicular to his midriff. As Oswald shone the light from his mobile on the body and beside where his right arm ended he spotted something written on the soft earth. The man had written something on the ground using the stone in his death throes. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">Oswald didn't know what to do as panic swept all over him he dash past and came to the spot where the nearby church is visible and with flailing arms and in English mixed with broken Swedish he told a few of the passerby's what he had see. He wasn't sure if the man was dead. By now few people had gathered around and people were passing opinions on what could be done. Someone from among the bystanders called up the polisen. As Oswald heard a siren in the vicinity he spotted that the dead man had probably bled from his head. There was thick dark fluid - coagulated blood - behind his head. It was clear he had died here and then Oswald spotted something scribbled on the ground. It was the famous bra ket notation used extensively in equations of Quantum Mechanics. He went closer and knelt down to see what it was when he realized that he had seen the same bra - ket notation elsewhere in the campus. Beside the notation were scribbled the words TV indistinctly. He had attempted to write something more but it was completely ineligible. </span><br />
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<i style="font-family: 'Trebuchet MS', sans-serif;">Wednesday, Polisen Station Lobby, Ernst Fontells plats, 8:00 PM </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">Chief inspector Jonas Einarsson was clean shaven and touched his forehead as he rested his elbow on the table, he sat in the chair opposite 'Tony' Oswald Maguire. </span><br />
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<span style="font-family: Trebuchet MS, sans-serif;">"So you think this thing that the man scribbled was a clue", said Jonas half smiling. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">Oswald nodded. "Yes", he said, " It seems to me he is indicating something, perhaps a location in the campus or an object". </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"What is this some sort of treasure hunt, you students are so messed up playing these games these days" said Einarsson waiving off the suggestion. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">That's when his phone rang and he spoke for a while. He came back and sat on his chair again. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"Do you know what it is?", said Einarsson, " A blunt weapon was used to kill the man, he was hit from behind, apparently his skull had cracked" he concluded. "Thanks for your cooperation, you can now take your leave", with this Einarsson rose, offering his hand for a handshake. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">Oswald was walking back home unnerved from the evening's happenings. He called up a friend of his and narrated what had happened and asked his friend to come over. </span></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">Thursday, Fysiken building, Chalmers Hogskolan, 12: 08 AM </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"This better be a clue", said Carl, Oswald's close friend as the two of them ran up the stairs. They had to get to the fourth floor of the building, to an open cafe. After Carl had come over to Oswald's place they had discussed about the case and had come to the conclusion that there was definitely a clue based on what the dead man had scribbled. There is an open cafe named the Branifiket which is the same bra-ket notation with the sign of the variable phi used to depict the Quantum wave function. This was on the fourth floor of the physics building which they were now racing towards. As they opened the door using the push button the long white corridor ahead lit up. They walked past and then entered an adjacent corridor that led to the open cafe. There were a series of large, framed photographs of Christopher Dalen the blind Swedish scientist and his inventions on the walls. As they neared the cafe - there is a wall in between the seating and the deli, someone bolted from behind the wall across the nearby door and ran across the opposite door to the glass bridge and disappeared. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">Oswald and Carl stood there flabbergasted and trembling, they didn't attempt to chase. They entered the cafe area and found behind the wall the counter now obviously shuttered at this unearthly hour. There were two dustbins and a TV on the wall opposite the deli. They felt behind the TV and peered at the space in between the TV and the wall where it was mounted. Finally Oswald came across a paper chit taped on top of the backside of the television. It was a piece of some notebook and it had a handwritten line stating: </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"If I Cullapse" </span></div>
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<b style="font-family: 'Trebuchet MS', sans-serif;">2. The Quantum Cipher </b></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">Saturday, Hogsbogatan, 10:30 AM </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">Melissa Stetson switched on the television and found the news channel blaring out that someone had been found murdered at the Chalmers University campus three days back. It was apparently a PhD student at Chalmers. His body had been identified and the weapon used was a blunt weapon with which he had been hit on the head. The polisen were further investigating the case. The suspect had not yet been identified and the person who had found the body was a Chalmers student Oswald Maguire. Melissa was a close friend of Oswald and immediately called him up. He gave her some details and asked her to come over to the study room near the branifiket cafe in the afternoon, he and Carl would be there. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">When she reached the studyroom she found Carl and Oswald huddled inside a study cubic. The place always made her cautious as the upholstery and sofas were all red. They were peering at the map of the University and there were a few sticky notes at different points on the map. It could have been any project she thought but this had to do with the murder on campus. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"We have found three clues till now" said Carl, after which they explained the details to Melissa. After that first clue led them to the cafe the second clue they had figured was a cipher clue Cu - stood for Copper which had led to the copper dome on the campus. There they had obtained a key and another piece of paper on top the entrance door which said: </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">'Last of the last Arch Enemy S of blue daffodils' </span><br />
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<span style="font-family: 'Trebuchet MS', sans-serif;">Melissa thought hard and felt this was again a location but for what? Then she suggested it might mean the southern archipelago of Gothenburg. This seemed possible. They would get there in the evening. </span></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">Saturday, Vrango islands, Southern archipelago Gothenburg, 4:30 PM </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">The ferry docked and the passengers all came out one by one. The two men came out last, one had a gash on his forehead from a recent injury and the other was over six feet tall, well built but not too hefty. They walked into the nearest cafe by the waterfront and ordered an espresso each. They chatted in Swedish and kept an eye on the outgoing ferry. </span></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">5:15 PM </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">The ferry Silvertarnan reached Vrango everyday at 5:15 PM. As the three of them got off from the ferry, the two men seated at the cafe turned to take a look. The description matched. One of the three was a stocky lad, with a muscular, tanned visage, the other was fair and lanky and the third was a girl and had a nose piercing passing through the center of the nostrils. It was time to see what the kids were up to. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"I have a feeling we need to walk to the end of the island as the clue says last of the last", said Oswald. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">They walked to the point where the island ended and now they had to find the house with blue daffodils. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"Do daffodils come in blue?", asked Carl to the two others. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"Maybe its not actually flowers", said Melissa. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">They walked around for some time at the edge of the island, where the craggy shoreline began were a series of houses mostly deserted. On one of the gates was a wooden sculpture of two blue daffodils and Melissa spotted it. They were about to open the door when Oswald waived "Stop let us wait". </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"I think some people are following us; lets walk towards the harbour on yth other side and lose the tail" </span></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">Saturday, Quasec Inc. - Facility D, Stockholm, 6:00 PM </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">The labs looked straight out of some spacecraft. The walls were different shades of orange and white. The Q logo in black was etched on the doors and on the ceilings. </span></div>
<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;"><br /></span>
<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">"We need the entanglement deconstructor device soon once the control tests are done, that will help us keep the collapsed wave intact and then we could do some tests in real", the man wearing the grey suit and a red cufflinks said. </span><br />
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Yes D'Silva wanted to keep that to himself, we got a few papers from his room at Chalmers but the details were not sufficient. Prof Dushyant too has no clue what D'Silva had figured out but we know for sure that he had reached a breakthrough", Tim Robbins the security expert for Quasec mentioned as he ran his fingers through his blond hair. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"No more clues on the device design? We need to get there before anybody else does", concluded the man in the grey suit with an air of finality. "Find out what Gunnar has to say", he said turning to face the third person in the room. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"They are in an island near Gothenburg right now, both Gunnar and another associate of his, we might be close to the device" said the wiry bespectacled man with greying hair in a raspy but polite voice. This is when his phone rang, he spoke for a while in Finnish and then hung up. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"The kids have found the device, Gunnarson had called - but we have a problem" he said. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"What problem?", asked the man in grey suit. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"Mr Huang, the kids had managed to escape with the device and Gunnar was only able to get the operating manual and the associated papers". </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"Find them and retrieve the device, now!", thundered Huang Kai Shek, his face turning a shade darker. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"Yes Sir" said the wiry man and walked out of the room dialing a number on his phone. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;"><br /></span></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">A few hours back </i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">The three of them hurried past the harbour and among the boats trying to loose the two people who were following them both in dark jackets and possibly carrying weapons. Why were they following them thy wondered. When the three of them came back to the house with the blue daffodils they had lost their followers or so they thought. When they opened the door with the key an entered they saw a clean room with just a chair a table and a bed. On the bed was asleep a young monk not more than fifteen years of age in a Nepali robe. The monk woke up due to the sound of the opening door. He stared at them calmly: </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"Do you stay here?", asked Carl. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">The monk nodded, and pointed to the table. There were a stack of papers in a folder which Melissa opened to check, it had some blueprints and a series of papers and mathematical equations worked out. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"What are these?", Melissa asked the monk. That's when the door came crashing down and there stood at the doorway two men in dark jackets revolvers in hand. Both wore Guy Fawkes masks.</span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"Hand it all over to us", shouted the short one. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"Is there a device or some instrument too? Find that now!", yelled the other person. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">Melissa handed over the folder and a book to Carl who in turn handed over the papers to the short man. </span></div>
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<span style="font-family: Trebuchet MS, sans-serif;">"We don’t have anything else", said Melissa. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"Is there anything else?", the tall man asked the young monk, who vigorously shook his head implying a no. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"If you get anything else let us know! And not a word to the polisen!!", said the short stocky scar-face handing over a piece of paper in Melissa's hand. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">"Or else …", he threatened with a gesture pointing his revolver in each of their directions. </span></div>
<div style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif;">The two men stormed out leaving Carl, Oswald, Melissa and the monk trembling. </span></div>
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<b style="font-family: 'Trebuchet MS', sans-serif;">3. Quantum wave constrictor </b></div>
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<i style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">Monday, Chalmers University, 12:30 PM </i><br />
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Imagine an hour glass, now think of the waist where it gets constricted. At that point the sand is forced to slow down and it flows down in a fine trickle. The same way reality can be seen as a wave where all possible realities or multiverses exist until the Quantum wave function of this moment collapses to create this moment.", said Professor Eckhart von Einhart. Professor Dushyant nodded in agreement. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"I understand that hypothesis but is there a way to constrict or control this flow is the question. When he was killed he was working on a condensate constriction implementation I believe as he had told me" </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Yes there is a possibility if there is enough density created at a point a mini Quantum wave constrictor can skrink the space-time fabric and the multiverses can be reduced to a single universe in the vicinity of the constriction. The constriction can be extended for short bursts or longer periods in space-time as time itself gets warped at that point. This something Satyendranath Bose had conjecture d in his last letter to Einstein" said Prof Eckhart with a smirk. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Wait, wait, wait. Are you telling me that this is like determinism for short bursts of time? I mean in other words controlling a snapshot of the foreseeable future? That's not possible in my humble opinion. The Schrodinger paradox would kick in making it impossible" </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"That doubt is valid. However a recent equation that your PhD student and another Chinese researcher had solved made this at least theoretically feasible if such a condensate can be created." </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Won't that take a massive research facility to create? What was it that got him killed. The police interrogated me; all I could tell them was that he was working on Quantum constriction" </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"An implementation is possible by means of a condensate coagulation but requires a high power source perhaps by means of a controlled nuclear fusion reactor of some sort" </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Don't forget the consciousness condensate too, a mind focused, I believe he was in touch with the Himalayan meditation foundation under Prof Brooks as they work on consciousness studies through meditation techniques" chipped in Prof Dushyant over his lunch. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Yes a source of consciousness is a necessary input to this process - a highly focussed and uncluttered source perhaps can be invaluable, what better than a monk trained in meditative practices!" said Prof Eckhart as he bit into his baguette with gusto. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Was he involved with a firm or any of the research groups working on such stuff? Might have been some inter firm rivalry or something confidential changing hands that got him killed. It is very unfortunate to have lost such a brilliant young mind at his prime", pondered Prof Dushyant to himself. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Yes very unfortunate", agreed Eckhart, he continued with "I have just heard of a handful of commercial firms working in this field, most others like DWave are into Quantum computing. In all I think two US firms, one in France and one in Israel although they may have subsidiaries in the Nordics" </span></div>
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<i style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">Quasec Facility D Stockholm, 13:05 PM </i><br />
<br />
<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">"So do we have coordinates yet for the component Z device? Without that the tests are null and void, we don't have working prototype of the comp Z based on the papers" </span><br />
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Should we also ask them to get the source too?", asked the scarfaced man. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Like I said without the constrictor device it won't work and abducting a source will only be a greater liability" </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"Do you think it was something related to this that got Dmitri bumped off?", asked the lady assistant. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">"We don't know for sure, let us focus on the task on hand", said Dr Schindler firmly. </span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif;">They moved into the Contrapt room as it was nicknamed. The room was white and at the center was a large tubular machine with several parts akin to stages of a rocket. There was a square-shaped glass box connected to the wall with some thick wires from the tubular contraption running to the end of the room where they seemed to connect to a large brown inlet possibly into another room. </span></div>
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<b style="font-family: 'Trebuchet MS', sans-serif;">4. The Treatise on Entanglement Networks</b></div>
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<i style="font-family: 'Trebuchet MS', sans-serif;">A few days later</i></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">'The world would be great if a mechanism were possible to choose among the different multiverses and decide which one you want to come true, but then could you switch back into one of other universes and change the course of the future again? Say a person has died and everyone knows he has died is it possible for that person to become alive based on a portal or a wormhole back into one of the universes in the multiverse where he is alive. Are there doors and portals from one universe into another within the entire multiverse? Universe networks? Are all the universes of the multiverse connected? That brings us to another premise the entanglement network.'</span><br />
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<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">'An entanglement network was a mechanism of entangling the state that one person was in with everyone else. Imagine I get to know that a particular X is dead and then I call others and show them the dead body and then the news spreads via television, phones etc and other communication mechanisms, so now the entanglement is complete all the minds are synchronized to know that X is dead (given that she really is dead)', read aloud Tony from on a diary entry in D'Silva's diary.</span><br />
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<span style="font-family: 'Trebuchet MS', sans-serif;">D'Silva had made the monk wear a helmet to create the quantum constriction effect, as the little monk told them. He had asked the monk to imagine a dead man hit on the head with a chisel and then his falling down and dying. He had told it was to be done exactly at a given time in the evening when a machine at Stockholm would receive his imaginary signal. Wasn't that exactly what Oswald had seen, but what about the message? Was it a way back into the multiverse network. Could you enter and kick-start another point of space-time in another universe of the multiverse, was it possible to bring someone back from the dead and make everyone change their minds?</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">'Was that then a form of disentanglement of the minds and a cross connection between universes - wouldn't that lead to too many combinations?', his diary continued.</span><br />
<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;"><br /></span>
<span style="font-family: 'Trebuchet MS', sans-serif; text-align: justify;">(Continued...)</span><br />
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-35426625631734744492015-08-06T15:27:00.002+05:302015-08-06T15:35:09.787+05:30Deepest Dreams<div dir="ltr" style="text-align: left;" trbidi="on">
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An Ode to Google #DeepDream - Perhaps generated by the same process - I see a scene in a park and start imagining other patterns ...</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitnpP1ysx1lW7yxxqhPWBfttzbirvkCNjHCVlafEpn34k_O038M1Cri2T73f40-z5Zn1kQoF_DZcXZpBJ47tqMd5RfcHCeqXozRcalIdXwKw4CeIuhAp8aoAehxqtvYk3KR_JS9w/s1600/dd.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitnpP1ysx1lW7yxxqhPWBfttzbirvkCNjHCVlafEpn34k_O038M1Cri2T73f40-z5Zn1kQoF_DZcXZpBJ47tqMd5RfcHCeqXozRcalIdXwKw4CeIuhAp8aoAehxqtvYk3KR_JS9w/s1600/dd.jpg" /></a></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-5464127596646391932015-03-28T05:14:00.003+05:302015-03-28T05:17:53.122+05:30Accepting Contradictions<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Georgia, Times New Roman, serif;"><b>Accepting Contradictions</b></span></div>
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<span style="font-family: Georgia, Times New Roman, serif;"><br /></span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">You want me </span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">To be like you.</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">And I want you </span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">To be like me.</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">Acceptance is hard, you see!</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">You won't change your mind,</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">And I agree with you too!</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">A space where it is impossible </span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">To change your mind - is a dead space. </span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">Bury it.</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">And yes </span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">It's OK to contradict yourself </span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">On a regular basis</span></div>
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<span style="font-family: Georgia, Times New Roman, serif;">Else you’re not even thinking ...</span></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-60913773112018934382015-03-25T13:24:00.001+05:302015-03-25T20:29:26.040+05:30BUS ROUTES & QUANTUM PHYSICS<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="line-height: 19.3199996948242px;">Was thinking about an analogy recently while getting back home from the tram station. </span><br />
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<span style="line-height: 19.3199996948242px;">Let us assume you are waiting at a bus stand for a particular bus. Quantum physics is akin to saying: Any given bus can be bus route number 5, 10 or 15 (or any other number) with any of the probabilities ranging from 0 to 1 at any time. It transforms into an exact bus number that arrives next when its probability of being that route number becomes 1. Or vice versa. </span><br />
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<span style="line-height: 19.3199996948242px;">But if you are on a particular bus the probability being the same route number is always 1. Or unless you have seen the bus yourself its route number is less than one but we cant be certain it might also be zero. </span><br />
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<span style="line-height: 19.3199996948242px;">Seems like a joke right? I know at the quantum (or very small scale) it maybe true, but can the analogy be used elsewhere in designing systems?</span></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-422427677723516232015-02-22T17:57:00.000+05:302015-02-22T21:09:51.992+05:30Topic modeling - Machine Learning<div dir="ltr" style="text-align: left;" trbidi="on">
Probabilistic <a href="https://dl.dropboxusercontent.com/u/38963516/ml2013/Blei2012.pdf" target="_blank">Topic modeling</a> provides methods for organizing, understanding, searching, and summarizing large electronic archives.<br />
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<b>Latent Dirichlet Allocation (LDA)</b>: The simple intuition behind LDA is that documents exhibit multiple topics.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRSJ10b48CetpYmYMTMAh3FxLKKZoaUX2m_NQqN8euWRS9RW3DttQzGBtbnoMrSdhMAQztcX32VNT9aZ_4loynXx7IPhMXH8ymyT_EMwsR59uR01FOpp9KEndFAwUUk9LCvf-fXw/s1600/Fullscreen+capture+22-02-2015+021903+PM.bmp.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRSJ10b48CetpYmYMTMAh3FxLKKZoaUX2m_NQqN8euWRS9RW3DttQzGBtbnoMrSdhMAQztcX32VNT9aZ_4loynXx7IPhMXH8ymyT_EMwsR59uR01FOpp9KEndFAwUUk9LCvf-fXw/s1600/Fullscreen+capture+22-02-2015+021903+PM.bmp.jpg" height="430" width="640" /></a></div>
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In reality, we only observe the documents, the other structure are hidden variables<br />
Our goal is to infer the hidden variables i.e. compute their distribution conditioned on the documents: <i>p(topics, proportions, assignments|documents)</i>.<br />
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<a href="http://psiexp.ss.uci.edu/research/papers/sciencetopics.pdf" target="_blank">Gibbs sampling for LDA</a> - Here we sample the topic of a word in one of the documents, given the topics of all other words, the topic distributions and the data. A sample from a recent coursework at Chalmers below:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMQX98vqPtJNRuzzLLuy7s1UvC-iGo8OHsfJJD74I0h8KxZHA_fXwElUXtHNm5KC9RsR0eLzQTlueNg43RFFcv9CSTV_WuqDLnTbOrngQOmBGNIRdT5WdHp31InBYfX34DHwbA9A/s1600/22_enlarged.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMQX98vqPtJNRuzzLLuy7s1UvC-iGo8OHsfJJD74I0h8KxZHA_fXwElUXtHNm5KC9RsR0eLzQTlueNg43RFFcv9CSTV_WuqDLnTbOrngQOmBGNIRdT5WdHp31InBYfX34DHwbA9A/s1600/22_enlarged.png" height="346" width="640" /></a></div>
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Another instance shown below - for 10 topics:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX3_bUeZyy1_LIwKDDvVPEYmlENzPpdEPQJhfjUuHDNJo3FNGL6qPWGgg8HTELqjPbtq7eM6anNo8TpAxkNFf2XjPEgCSdb3N9i2vwQ2vhstXbgv79a1uIeyryxs6qzXcrpUp3aw/s1600/22_2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX3_bUeZyy1_LIwKDDvVPEYmlENzPpdEPQJhfjUuHDNJo3FNGL6qPWGgg8HTELqjPbtq7eM6anNo8TpAxkNFf2XjPEgCSdb3N9i2vwQ2vhstXbgv79a1uIeyryxs6qzXcrpUp3aw/s1600/22_2.png" height="480" width="640" /></a></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-80594627383285688002015-01-28T00:19:00.001+05:302015-02-28T15:28:18.953+05:30Introduction to Population Genetics - Basics<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Georgia, Times New Roman, serif;">Population genetics is concerned with the study of the genetic composition of populations. This composition may be changed by segregation, selection, mutation, recombination, mating structure, migration, and other genetic, ecological, and evolutionary factors. Therefore, in population genetics these mechanisms and their interactions and evolutionary consequences are investigated.</span><br />
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<span style="font-family: Georgia, Times New Roman, serif;">A gene may have different forms, called alleles.</span><br />
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<span style="font-family: Georgia, Times New Roman, serif;">The allelic composition is called the genotype, and the set of observable properties derived from the genotype is called the phenotype.</span><br />
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<span style="font-family: Georgia, Times New Roman, serif;">Thus, supposing that there are two alleles A1 and A2, there are three possible genotypes, A1A1, A1A2, and A2A2. In the first and third case, the organism's genotype is homozygous (for A1 or A2, respectively), in the second case it is heterozygous.</span><br />
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<span style="font-family: Georgia, Times New Roman, serif;">Genetic material is deoxyribonucleic acid (DNA). It consists of four bases: adenine (A), guanine (G), thymine (T), and cytosine (C). Each base is linked to a sugar and a phosphate group, yielding a nucleotide.</span><br />
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<span style="font-family: Georgia, Times New Roman, serif;">The nucleotides are arranged along two chains to form a double-stranded helix in which the pairings A-T and G-C between the strands are formed. Therefore, all the genetic information is contained in each of the two strands.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJazGHmlLrpdxCw1_y4-gFRTyYRT-87Z30FXxVkeNoxwHNwbeGO3gbO2twUWf6mw7lWOZK74piym4KaTcdmqsDVShENRL1xTiDReA-Mt1wX_oqd2hOVUs3YzeZUDJj8ARyaZV1sA/s1600/DNA_Structure.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJazGHmlLrpdxCw1_y4-gFRTyYRT-87Z30FXxVkeNoxwHNwbeGO3gbO2twUWf6mw7lWOZK74piym4KaTcdmqsDVShENRL1xTiDReA-Mt1wX_oqd2hOVUs3YzeZUDJj8ARyaZV1sA/s1600/DNA_Structure.png" height="312" width="320" /></a></div>
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<span style="font-family: Georgia, Times New Roman, serif;">Three bases code for one amino acid, which are the building blocks of polypeptide chains and proteins.</span><br />
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<span style="font-family: Georgia, Times New Roman, serif;">A gene typically represents a contiguous region of DNA coding for one polypeptide chain. Its position along the DNA is called the locus, and a particular sequence there is called an allele.</span></div>
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shamitbagchihttp://www.blogger.com/profile/03538037501379860935noreply@blogger.com0tag:blogger.com,1999:blog-3759549.post-23681789212901966872015-01-24T03:55:00.003+05:302015-01-28T00:06:37.248+05:30Jai Hind, Jai Hindi!<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwGyU5mvE7ui3xH7RJwMC4yFVHZI8IYd2PTLFtIA0tnWv1JBkFxKflFfoIKABD5mfuMTSyuJGiz1DVr5tNBbpVgRndMyNnWu1efaEiUKroqUQz9cTQIcJy4HneFl08vKpFtT9VRg/s1600/Fullscreen+capture+23-01-2015+073449+PM.bmp.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwGyU5mvE7ui3xH7RJwMC4yFVHZI8IYd2PTLFtIA0tnWv1JBkFxKflFfoIKABD5mfuMTSyuJGiz1DVr5tNBbpVgRndMyNnWu1efaEiUKroqUQz9cTQIcJy4HneFl08vKpFtT9VRg/s1600/Fullscreen+capture+23-01-2015+073449+PM.bmp.jpg" height="338" width="640" /></a></div>
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Dear Mr Bagchi,<u></u><u></u></div>
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We have received your entry for the World Hindi Day, which has been selected for an appropriate award.<u></u><u></u></div>
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Please confirm your attendance on the flag hoisting ceremony on 26<sup>th</sup> January, 2015 at 0915 hrs at India House 13-A, Villagatan, Stockholm to receive the prize.<u></u><u></u></div>
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Regards,<u></u><u></u></div>
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