Post AnvskShffDMrSLxffs by mmby@mastodon.social
(DIR) More posts by mmby@mastodon.social
(DIR) Post #Anvp50PJfBWXxiutDU by futurebird@sauropods.win
2024-11-11T11:26:23Z
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It *shouldn’t* surprise me that a continuous one-to-one function must be strictly increasing (or decreasing.) AND YET every time I return to analysis I’m shocked anew by this simple fact. Maybe I’m easily impressed? No, I think what this really shows is that continuity is a much more complex concept than it at first appears. Non-mathematicians have a folk understanding of continuity: but it is precisely those more widely useful mathematical concepts that are the most deceptively simple and rich.
(DIR) Post #AnvqPBWa8hhCBgV8PA by Scmbradley@mathstodon.xyz
2024-11-11T11:41:22Z
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@futurebird there's a nice history of the concept of continuity in Imre Lakatos' "Proofs and Refutations" (I think it's appendix c?) where he goes through all the attempts to define it properly and all the pathological functions that demonstrated a need for a more careful definition. It's neat.
(DIR) Post #AnvqVMDhHrA561bxdw by llewelly@sauropods.win
2024-11-11T11:42:31Z
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@futurebird I think a one-to-one function requires a strict total ordering of both the range and the domain?
(DIR) Post #AnvrGLbVOY3knQ9bg8 by benni@social.tchncs.de
2024-11-11T11:51:00Z
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@futurebird i was mindblown by this fact as i realized it holds in topological spaces with there completely different definition for continuity, too.
(DIR) Post #AnvskShffDMrSLxffs by mmby@mastodon.social
2024-11-11T12:07:38Z
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@futurebird a common 'trick' in ray tracing for sampling arbitrary probability distribution functions relies on thatyou use the cumulative distribution function, which is usually strictly increasing and invert it for uniform samplinghttps://en.wikipedia.org/wiki/Inverse_transform_samplingwhen I first encountered that I also felt a sort of surprise at it
(DIR) Post #Anvu4BpKDhDGrPVpa4 by llewelly@sauropods.win
2024-11-11T11:58:04Z
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@BenAveling @futurebird ah, yeahbut what does continuous mean if the range or the domain is , say, the integers?
(DIR) Post #Anvu4CtcFEEuB0WluK by futurebird@sauropods.win
2024-11-11T12:22:25Z
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@llewelly @BenAveling If you have only the integers wouldn’t they be adjacent incrementally?
(DIR) Post #AnvuC5WTXfHhcqShEG by futurebird@sauropods.win
2024-11-11T12:23:52Z
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@benni “if you turn around you are retracing your steps” doesn’t seem that deep— but it’s kind of amazing.
(DIR) Post #AnvufavXrP67QFG2jY by futurebird@sauropods.win
2024-11-11T12:28:56Z
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@eirias Yes. But to me that’s the simpler concept. Though one to one functions that aren’t continuous are VERY interesting everything gets mapped somewhere every element in the set has a soulmate. Ok maybe it’s just as complex.
(DIR) Post #AnvupE4NhNtPQWz1tI by futurebird@sauropods.win
2024-11-11T12:30:51Z
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@frank I thought “one-to-one” was the less technical phrase? though isn’t “bijection” the better match?
(DIR) Post #AnvvbTKQGovMeQUZeK by llewelly@sauropods.win
2024-11-11T12:39:39Z
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@futurebird @BenAveling I think so? I guess adjacency is necessary but not sufficient for continuity?
(DIR) Post #AnvxEoW1WyS7brF3Wy by chrisamaphone@hci.social
2024-11-11T12:57:57Z
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@frank @futurebird yeah one to one means bijective. the “less technical” terms I learned for these wereinjective = intosurjective = ontobijective = one-to-one“onto” for surjective makes sense to me but the other two seem more arbitrary
(DIR) Post #AnvzGbPzOgefTbOrOi by IngaLovinde@embracing.space
2024-11-11T13:20:39Z
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@futurebird hmm it's not exactly obvious but seems to be quite easy to prove?
(DIR) Post #Anwsnsbm06wZpFnC8O by soaproot@sfba.social
2024-11-11T23:42:59Z
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@futurebird Those who know more mathematics than me assure me that continuity is indeed very rich. This has been especially true as I've learned about topology and #constructiveMathematics (both of which feature continuity in a central way).
(DIR) Post #AnxtkPCBC85QqZgjnE by futurebird@sauropods.win
2024-11-12T11:28:16Z
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@eirias If —if love is continuous and if everyone has a soul mate then isn’t it true that at least one person must be a fixed point?(though I think it’s not sensible to talk about a continuous function on a discrete set?)
(DIR) Post #Anxub7YVVfjdUBGiJ6 by todymotmot@mastodon.social
2024-11-12T11:37:47Z
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@futurebird @eirias Ifif love is continuous,And if everyone has a soul mate,Then isn't it trueThat at least one personMust be a fixed point? Though I think It is not sensibleTo talk about A continuous function On a discrete setMorning poetry by myrmepropagandist 🥰
(DIR) Post #AnxucpPzuluZoU6PlA by futurebird@sauropods.win
2024-11-12T11:38:07Z
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@eirias *weeping*“Dirty cheating dogs like, you, Ahmed— you are the reason LOVE is NOT a function!!!”(sad math tales)
(DIR) Post #AnxukeAVcohVEtaOS8 by ekis@mastodon.social
2024-11-12T11:39:30Z
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@futurebird @eirias My love is recursive
(DIR) Post #Any2dsHLZ2uibk49FQ by BashStKid@mastodon.online
2024-11-12T13:07:55Z
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@futurebird @eirias Perhaps love is both complex and a transformation?
(DIR) Post #AnyAi81TllNyEFBQcC by jmax@mastodon.social
2024-11-12T14:38:20Z
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@futurebird @eirias On a discrete set, you simply have to check at each set member. Which is the kind of thing that gets you a reputation.