A pointed topological space is a pair [math]\left( {X,x} \right)[/math] consisting of a topological space [math]X[/math] and a choice of basepoint [math]x \in X[/math].A morphism of pointed spaces is a continuous map preserving basepoints.If [math]\operatorname{Top} [/math] is the usual category of topological spaces, then the category of pointed spaces (denoted [math]{\operatorname{Top} _*}[/math]) can be constructed as the undercategory [math]{}^{*/}\operatorname{Top} [/math] of the single-point topological space.The category of pointed spaces has products and coproducts as follows...The product is the cartesian product given in the "obvious" way.The coproduct is the wedge sum [math]X \vee Y[/math] given as a pushout [math]\begin{array}{*{20}{c}} *& \to &X \\ \downarrow &{}& \downarrow \\ Y& \to &{X \vee Y} \end{array}[/math]There is also a functor [math] \wedge :{\operatorname{Top} _*} \times {\operatorname{Top} _*} \to {\operatorname{Top} _*}[/math] called the smash product.It is given as a pushout [math]\begin{array}{*{20}{c}}Comment too long. Click here to view the full text.
>>9355388Judging from this I reckon you are close to getting a PhD
>>9355092An [math]{\mathbb{E}_\infty }[/math] - ring [math]A[/math] is called connective if the homotopy groups vanish for negative degree.(note all categorical things discussed in the following are infinity-categorical)We define the category of affine spectral schemes [math]\operatorname{SAff} [/math] as the opposite category of the category [math]\operatorname{Rg} \operatorname{Sp} [/math] of [math]{\mathbb{E}_\infty }[/math] - rings.[math]{\operatorname{SAff} _{ \geqslant 0}}[/math] , the category of connective affine spectral schemes, is defined as the opposite category of connective [math]{\mathbb{E}_\infty }[/math] - rings.We denote the functor by [math]{\mathbf{Spec}}:\operatorname{Rg} {\operatorname{Sp} ^{op}} \to \operatorname{SAff} [/math] .Explicitly, an affine spectral scheme [math]{\mathbf{Spec}}\left( A \right)[/math] is a pair consisting of an affine scheme [math]X = \operatorname{Spec} \left( {{\pi _0}\left( A \right)} \right)[/math] and a sheaf [math]{\mathcal{O}_X}:\operatorname{Open} {\left( X \right)^{op}} \to \operatorname{Rg} \operatorname{Sp} [/math] of [math]{\mathbb{E}_\infty }[/math] - rings.
>>9355638A morphism [math]f:A \to B [/math] of [math]{\mathbb{E}_\infty }[/math] -rings is called etale if the induced morphism of commutative rings, [math]{\pi _0}f:{\pi _0}A \to {\pi _0}B [/math] , is etale.A morphism [math]f:A \to B [/math] of [math]{\mathbb{E}_\infty }[/math] -rings is said to exhibit [math]B[/math] as a localization of [math]A[/math] at [math]a \in {\pi _0}A[/math] if (i) [math]f[/math] is etale(ii) [math]f[/math] induces an isomorphism [math]\left( {{\pi _0}A} \right)\left[ {{a^{ - 1}}} \right] \cong {\pi _0}B [/math]In this case we denote [math]B \equiv A\left[ {{a^{ - 1}}} \right][/math] .-------------------------------------------------------------------------------------------------------------------Let [math]A[/math] be an [math]{\mathbb{E}_\infty }[/math] -ring. Then the structure sheaf [math]{\mathcal{O}_X}[/math] of [math]X = {\mathbf{Spec}}\left( A \right)[/math] satisfies the following conditions:(a) For all [math]a \in {\pi _0}A[/math] define [math]{\operatorname{U} _a} = \left\{ {p \in \operatorname{Spec} \left( {{\pi _0}\left( A \right)} \right)|a \notin p} \right\}[/math] , the usual basis for the topology of affine scheme. Then there is an equivalence of an [math]{\mathbb{E}_\infty }[/math] -rings, [math]{\mathcal{O}_X}\left( {{\operatorname{U} _a}} \right) \cong A\left[ {{a^{ - 1}}} \right][/math] .(b) [math]{\pi _0}{\mathcal{O}_X}[/math] is the usual structure sheaf of the affine scheme [math]\operatorname{Spec} \left( {{\pi _0}\left( A \right)} \right)[/math] .
>>9355638>>9355765To clarify, a Sheaf of [math]{\mathbb{E}_\infty }[/math] -rings is a contravariant (infinity) functor [math]\mathcal{F}:\operatorname{Open} {\left( X \right)^{op}} \to \operatorname{Rg} \operatorname{Sp} [/math] from the category of open sets on the topological space [math]X[/math] to the category of [math]{\mathbb{E}_\infty }[/math] -rings, such that the functor satisfies a descent condition for the topology on [math]X[/math].The derived structure of these spectral schemes is coming from the fact the structure sheaves have nontrivial higher homotopy.The nth homotopy sheaf of [math]\mathbb{F}[/math] is the sheafification of the presheaf [math]\left( {{\pi _n}\mathcal{F}} \right)\left( \operatorname{U} \right) = {\pi _n}\left( {\mathcal{F}\left( \operatorname{U} \right)} \right)[/math] .For an affine spectral scheme (and in fact a general spectral scheme that I won't define) the higher homotopy sheaves [math]{\pi _n}{\mathcal{O}_X}[/math] are quasi-coherent [math]{\pi _0}{\mathcal{O}_X}[/math] -modules.Moreover we have isomorphisms [math]\left( {{\pi _n}{\mathcal{O}_X}} \right)\left( X \right) \cong {H^{ - n}}\left( {X,{\pi _0}{\mathcal{O}_X}} \right)[/math] .
>>9355840>Fshould be [math]\mathcal{F}[/math]
Why wouldn't some form of life happen on planets around Earth? It seems like a lot of crazy shit for it to happen in a vacuum (hey-o!).
according to my cipher... pic related meansUNITEDARABEMIRATESHAVEURANIUMACTUATEDHEMORRHOIDAMPUTATING EQUIPMENT
How to trigger feather-fags.
>one shot at life>born a brainlet
Be a biologist then.
>>9355937>one shot at life>born in the third world
I'd rather be attractive than smart
How do you construct the set of real numbers from integers?
I like it like this:-Field-Ordered Field-Supreme AxiomBut I guess you can use Dedekind for a simple, conceptual approach.
>>93525832^N
>>9353578An uncountably infinite set of sequences over {0,1}
>>9352583You don't. Take the axiomatic approach instead.Every Dedekind-complete ordered field is isomorphic to the real numbers. Hence the real numbers are *the* Dedekind-complete ordered field.
This is why nobody takes set theory seriously
Choose your fighter /sci/
>>9355612Well when you make the wizard seem like the obvious choice, that's the one I'm gonna choose
How many points did you get on your last Putnam competition? You do take part in those, right?
Took my first one a few weeks ago. I am sure I got at around 10-20 pts
>>9355334I got 10 the last time I wrote it.
I have 2 questions about evolution:1- how does it explain creatures like pic related?2- if life exists for another billion years but still remain 100% the same on a macro level, would it disprove evolution?
>>9355711Thank you. They look cool.
>>9355685>Richard DawkinsHe's a moron making money off of angsty teenage atheists.
The 'how' of macroscopic structure change is always an interesting one, as there are many molecular pathways that are taken. However, if you're talking about simple evolution by allele change, a classic example is to look at the curious case of beak length of birds in tropical islands that are roughly insulated. Depending on the climate and weather, birds of the same species but differing beak size will be more successful, with longer ones being able to reach further into trees and the like. If you can imagine a case where it was so much more advantageous for the long beaked birds to survive, then a scenario where the tiny beaked ones don't even live til reproduction becomes a natural conclusion.Going back to how the differences or mutations arise, it's a well-documented molecular phenomenon. In the process of gamete creation, or making the sperm/egg, there's a cellular process called meiosis where DNA recombination occurs. I suppose I'll make a second post going into that and explaining exactly HOW an error might happen, but errors do happen, which are actually fairly simple to recognize.1) A segment of DNA is replicated.2) A segment of DNA is inverted- essentially turned so that it's the opposite end first when put back into sequence.3) A segment is deleted4) A single nucleotide is changedAnd then a sort of a weird one5) Transposition/retroviral mechanism occurs.All of those add up over time to create new features, which then through sheer chance might end up with some 'function'. As soon as a feature becomes a function, then it can undergo pressure-based selection.The simplistic nature of how there are many pathways to reach some kind of effect is demonstrated well on the molecular level by this. (Pic attached), The way the active site of hemoglobin and chlorophyll are similar in chemical construction lends itself to the notion of radically different origins for similar processes, which is an incredibly interesting field of study.
>>9355855To add;So, a single DNA pair strand coiled up is called a chromosome. You may know that we all have 23 unique sequence pairs of chromosomes, but that's actually 46, with two copies of each. In the attached picture is both phases of meiosis. What happens is that each chromosome duplicates itself- and this is usually a perfect copy, but this stage itself could replicate imperfectly, usually leading to issues like 3 or 4, where some part is missing, or a single nucleotide is incorrectly put in place. You have probably seen chromosomes look like two sort of stubby looking appendages and then one area where they get narrow and connect, and that narrow area is the centromere- it's just sort of a non-coding mechanistic region of DNA that a protein complex called the kinetocore latches on to. In the first action packed stage of Meiosis, the pairs that are being split are the non-identical mom and dad versions, not the identically copied ones (those are for part 2).Now, at the opposite poles of a cell, are things called 'spindles', which have microtubules, like tenticles that lash around at this stage of meiosis. Once they latch on to the kinetocore, they tug a bit. If the kinetocore undergoes tension, it stops releasing a signal that stops the cell from moving through meiosis by breaking down the cohesins (basically molecular glue) between the DNA pairs. In meiosis 1, the tension across the kinetocore is created by actually physically breaking the DNA of all 4 strands, and then fusing them all together. It creates bridges, called chiasma, which will then make them intertwined so the tension is generated. Once the cohesins are gone, the DNA is violently ripped, leaving some strands with sections from the other, and offering plenty of opportunities when the DNA is being repaired to either flip around, go missing, change nucleotide, etc.Meosis 2 is boring, it's just the separation of the 2 copies, so you end up with sperm/egg w/1 copy of each chromosome.
>>9355865God I'm retarded, I didn't add the picture. At any rate, these mutations occur at different rates dependent on the species- some species have protein variants that are much more delicate than their homologs in other species, which will then create fewer mutations. Some purposefully have more mutations because they are in environments that require the addition of new variants that can then be selected for. It's on a weird sort of meta level that's hard to examine- after all, evolution doesn't plan for the future, but faster adapting lineages will often have greater reproductive success.To talk about OP's example of the bug which looks like a leaf, let's think of some novel example of a bug, let's say a cockroach or any other model bug. In one generation, there's a crapshoot of offspring, one of which has a mutation in the gene that controls for pigment that causes the pigment to not be expressed as much because it's a broken protein. Then, instead of being dark brown, the bugs with this mutation are light brown. They are more successful because the animals that eat the bugs can't see them as well against the trees. Then, let's say that there's a few amino acid changes to a pigment protein that, when they do their biomolecular production, they end up synthesizing molecules that contain different resonant patterns than the one it's used to, and it absorbs a different wavelength of visible light- this time, reflecting green. Those have such a higher chance of survival that eventually, all of the ones that are brown can't compete- of course they would have reproductive success at first, but if each generation they end up not being able to get the resources because they're outcompeted by the green ones, their lineage will die off. From there, the rest of the structures that are leaf-like aren't too difficult to imagine.
hi /sci/Could you guys give hints on any of these? Studying for a final and would appreciate some help
Is english the most efficient language in the world?
>>9350022Sanskrit.
>>9352776>>9353549>>9350109>>9354243Ithkuil anon here... Those aren't words with arbitrary meanings. The language is made up of multiple affixes that each together. All of these affix reflect grammatical categories that are actually found in various natural languages as well. For example:Edhál - Water for nutritionAidhawél - Drink water!Aidhás qo - He is drinking waterM-mraidhás qo - He will be drinking waterM-mraidhád qo - He will have drank waterLam-mraidhád qo - It is a fair guess that he will have drank waterLam-mraidhád oi qo - It is a fair guess that he will have drank the water all in one gulpLam-mraidhadá'kš oi qo - Damn it, it is a fair guess that he will have drank the water all in one gulp.That is just an example I just created off the top of my head, but yeah, it isn't random. Something to note however is the languages goal isn't simply efficiencyAnyways, thank you. I will be shilling every week for $0.05 an hour. For questions and concerns, please contact me at [email protected]. Lojbanists are NOT invited, logbarian scum.
>>9355618Well, I guess my Ithkuil is better than my English!>*The language is made up of multiple affixes that each work together. All of these affixes
>>9350022Probably not. English has heavily simplified over time due to its rapid spread. It's really a victim of its own success. More complicate but more efficient languages would develop when a society has been relatively isolated for a long amount of time.
>>9350022pretty much any linguist will say that all languages are equal and all have the same capabilities as each other. in other words, no, english is not the most efficient language in the world, nor is chinese, spanish or icelandic simply because they all do the same thing with varying mechanics, functions and sounds. any idea that can be thought of can be expressed, no matter what language you speak
Would a 5th generation downie still be retarded?
>>9355503>>9355446>>9355473If they have 3 chromosomes I'd imagine they could not but they could if it was due to a translocation or two of the chromosomes were combined.
>>9355503I guess it's possible, but apparently many of them (almost all males and many females) are sterile. If I had to guess, those that are are able to have kids, and do have healthy children probably created gametes with the normal number of chromosomes.Here's an interesting report:http://www.jaypeejournals.com/eJournals/ShowText.aspx?AID=16&ID=764&IID=70&IN=_eJournals%2FInternational+Journal+of+Infertility+and+Fetal+Medicine.jpg&TYP=TOP&Type=FREE&Year=2010&isPDF=YES
>>9355439>Would a 5th generation downie still be retarded?Pewdiepie is retarded so I think we can probably assume it's true.
>>9355553>Making millions in a month>Retarded
>>9355841>successful => non-retarded
aint it the tooth.
>>9351344I take it you haven't taken a single physics or chemistry class at college level in your life.
>>9351344PhysicsNo abstractionsIt's like you didn't even take physics 2.
>muh proofs>muh shit job>muh cuck majorI minored in physics and I swear that major is full of the biggest autists I've ever seen. The levels of mouthbreathing were unreal
>>9355593Not to mention Einstein is a fraud and the aether is real.
>>9355633hell yea boi
>met engineer>tell him I am in intro to linear algebra>he chuckles and says he is in linear algebra the more advanced one>I explain his linear algebra is the engineer version and intro is the one with all the proofs and theories>he chuckles and says I'm wrong>we exchange midterms>I get 100% on his in 35 minutes>after an hour he gives up with a 24%
>>9353519this>>9353580probably true but same could be said of "programming for mathematicians and scientists" and basically any class that has "for" in the title
The state of murrican education: brainlet engineers are allowed to take math classes without proofs.
woah cool blogwhere do i unsubscribe tho? ?
>>9353485DAE ENGINEERS ARE USELESS MATH MAJORS FTW XDDDD
>>9354619Why the homophobia?
Anybody else watching the new season?
>>9355074I'm in a party right now, I asked the nearest people to me about the problem, none of them knew about it, your argument apart from irrelevant seems false.
>>9355303>partynormies out, reee etc.
>>9355312I didn't go because I wanted to, it was boring as shit.Im not a normie anon.
>>9354853Nah, waiting for the episodes to come out on piratebay.What? You think i'll pay for this shit?
>>9354853God damn it's the exact same garbage as the first season. Overproduced filler that repeats the same message over and over and over again to somehow turn a 10 minute video into a half hour one. Absolutely annoying and borderline intolerable to watch. Wish he'd just get to the fucking point and fill in the remainder of the episode's timespan with more content.
Can anyone tell me if I'm going to be late to the party when it comes to the breakthroughs in artificial intelligence? I'm currently doing my bachelor in CompSci and I'm planing to do my masters as well and get a PhD in machine learning. I would be done with that in 8 - 10 years. Am I going to be late? That AlphaGo event got me anxious we will achieve things earlier than we expected
>>9354024>Corporate event to hype sponsors got me anxiouskys soyboy
>>9354024(y) >>9354793(n)
>>9354024>I'm currently doing my bachelor in CompSci and I'm planing to do my masters as well and get a PhD in machine learningWhat the fuck are you doing a bachelor in CompSci? Are you retarded?
>>9355742What's wrong with a bachelor's in compsci?
>>9355801
