2026-06-04T08:39:00Z https://yabu.me Ethan MacBrough https://yabu.me/npub178a94wsk5sexgxa02zyrp9us7l3upzlnuzt4ulp72q3als5snjts2usvwx https://media.mathstodon.xyz/accounts/avatars/109/497/082/895/447/284/original/bb9d3b05c45d5953.jpg https://media.mathstodon.xyz/accounts/avatars/109/497/082/895/447/284/original/bb9d3b05c45d5953.jpg https://yabu.me/nevent1qqstkkr43v0mtu0lx5zpy4a4639qzl8ge7pjasks30grk3jhmdrm30qzyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfwrrktzc In reply to <a href='/nevent1qqsrsp5pfpgahfwfdz4yd66sdn9hxq06ufwyw65de356qys96z4wx3g42yt58'>nevent1q…yt58</a><br/>_________________________<br/><br/>Oh, I see, I forgot the state w is supposed to be a linear combination of all permutations of RGB when moving from the first post to the second. Thanks for the clarification. 2025-02-10T19:51:08Z https://yabu.me/nevent1qqsrre7r5q63499rdpcqh9fg9t3adepyxs7srdqez7h8yu26jah9txczyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfw9f724s In reply to <a href='/nevent1qqst2khf2dhkxu36c4qhqwp2atuvqldvql0f8nj7lhtkyzghcvyzues2h4cx4'>nevent1q…4cx4</a><br/>_________________________<br/><br/>Thanks for the post, I found it very enlightening. But I&#39;m having trouble understanding<br/><br/>&#34;The vector ... will contain a component in the RRR direction of U_{11} U_{12} U_{13}, but all six terms will only differ by a sign, and the sum will be zero.&#34;<br/><br/>Isn&#39;t the U_{11} U_{12} U_{13} term the only thing contributing to the RRR coefficient? What are the six terms you reference here? 2025-02-10T17:22:37Z https://yabu.me/nevent1qqswwd25e308ypmzkd5j78x7cpv7uxhzhz757hrycl3ddy26fsp5wlszyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfw0k5nzk In reply to <a href='/nevent1qqsp4trk4fmmplt6m9em4ajasc2yuqnpc6e76qh54v7hzqshcwj8fzqnzhj64'>nevent1q…hj64</a><br/>_________________________<br/><br/>No, I&#39;m the one who is wrong. I guess what I was calling finite-type is typically called &#34;locally finite-type&#34;, and finite-type = locally finite-type + quasi-compact. So indeed finite-type implies finitely many points for schemes.<br/><br/>Also I figured out how to reduce the algebraic space claim to the scheme claim:<br/><br/>By definition we have a surjective étale map phi: U -&gt;&gt; X with U a scheme. To show desired finiteness for X, it suffices to show U is finite type. Since étale maps are locally finite-type, we know U is locally finite-type. Now let U_i be an open cover of U by finite-type schemes. Since étale maps are open we obtain an open cover of X by phi(U_i)&#39;s. Since X is quasi-compact we cover X by phi(U_1) ... phi(U_n). Then the restriction of phi to U&#39; := U_1 cup ... cup U_n gives a surjective étale cover of X by U&#39;, and U&#39; is finite-type by construction. 2025-02-05T01:07:37Z https://yabu.me/nevent1qqszcmzu982wckpnzvr9c9qxhmvw7m9x9e4272prdn0hq24qwvnlfhczyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfw7wl5dl In reply to <a href='/nevent1qqstn3sw65y8c5ugnespykctsz04mcxs45v39cdwyggseph5lfcepycxdfv08'>nevent1q…fv08</a><br/>_________________________<br/><br/>I don&#39;t think that really fixes the example, just take a disjoint union of arbitrarily many Spec(Z)&#39;s. 2025-02-04T21:29:57Z https://yabu.me/nevent1qqsvz25putxxdz4cee049yvfeudqgnp99ykpl2jvwgnm39zr94zu0qczyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfwstxm9c In reply to <a href='/nevent1qqsgtfn067c9s5zg55h9pvq4nj0hj8kvpyup9ta0srhquds47pgdlqsa9qxzl'>nevent1q…qxzl</a><br/>_________________________<br/><br/>Actually I can&#39;t figure out how to ensure quasi-compactness pulls back to U. I think you might need to directly show that there exists some étale cover U -&gt;&gt; X with U a quasi-compact scheme. 2025-02-04T18:20:59Z https://yabu.me/nevent1qqsgtfn067c9s5zg55h9pvq4nj0hj8kvpyup9ta0srhquds47pgdlqszyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfwjj2m57 In reply to <a href='/nevent1qqspcx20txnzn800f99lyema6mp86g8ela3mj4rztd6ure4d5qk6ykc4285j3'>nevent1q…85j3</a><br/>_________________________<br/><br/>As stated this isn&#39;t even true for schemes, e.g. the disjoint union of arbitrarily many Spec(k)&#39;s is finite-type over k. You need to add some topological finiteness criterion like quasi-compact.<br/><br/>With this additional assumption your claim definitely holds for schemes (cover by finitely many affines, each of which embeds in some A^n), and it should also hold true for algebraic spaces: by definition you have an étale covering U -&gt;&gt; X with U a scheme, and then adjective calculus implies U will also be finite-type and quasi-compact, so you can reduce to the scheme case. 2025-02-04T17:48:09Z https://yabu.me/nevent1qqs9pd8cm2x8mwxg27nezk5q05ekmlxewgzzn2uwc6rlkhxf5ha8fgczyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfwgkkau0 In reply to <a href='/nevent1qqs9c0ewtu39mstx80rztwakvhqeeerhqm0hsgwtm5ytpm0mudmsqtgpkh32e'>nevent1q…h32e</a><br/>_________________________<br/><br/>Ah, got it! For some reason when I read &#34;the liquid in the raindrops affects the size of the rainbow&#34; my brain jumbled it up and I came away thinking the size of the raindrop affects the rainbow. Oops! 2024-09-15T20:07:56Z https://yabu.me/nevent1qqs8kc7sgmlsa2874k9x0jeywkdrnmqcmgfpcac43jxrp7pepmaqxlgzyrcl5k46z6jryeqm4aggsvyhjrm78syt70sfwhnu8egz8h7zjzwfwe8562e In reply to <a href='/nevent1qqsdmq8ncppr6kntyr5z00jpgxy7cu2fgnkwvdp52wtp7ywwe5a4qhshk8nwv'>nevent1q…8nwv</a><br/>_________________________<br/><br/>Does this account for the size difference of (m)ethane raindrops vs water raindrops?<br/><br/>Naively I would expect (m)ethane raindrops should be larger due to lower surface tension (although maybe this is counteracted by higher air pressure?), which might counteract the lower refractive index. 2024-09-15T19:22:45Z