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| 2026-07-06 23:41:15 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) (Ping timeout: 245 seconds) |
| 2026-07-06 23:40:07 +0000 | emilym | (~Thunderbi@user/emilym) (Ping timeout: 272 seconds) |
| 2026-07-06 23:39:10 +0000 | merijn | (~merijn@62.45.136.136) (Ping timeout: 254 seconds) |
| 2026-07-06 23:38:47 +0000 | machinedgod | (~machinedg@d108-173-95-19.abhsia.telus.net) machinedgod |
| 2026-07-06 23:38:19 +0000 | czan | (~czan@user/mange) czan |
| 2026-07-06 23:36:58 +0000 | skum | (~skum@user/skum) skum |
| 2026-07-06 23:35:31 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) |
| 2026-07-06 23:35:16 +0000 | emilym | (~Thunderbi@user/emilym) emilym |
| 2026-07-06 23:34:44 +0000 | czan | (~czan@user/mange) (Quit: Be back eventually!) |
| 2026-07-06 23:32:17 +0000 | merijn | (~merijn@62.45.136.136) merijn |
| 2026-07-06 23:24:16 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) (Ping timeout: 248 seconds) |
| 2026-07-06 23:23:29 +0000 | acidjnk_new | acidjnk |
| 2026-07-06 23:23:23 +0000 | lisbeths | (uid135845@id-135845.lymington.irccloud.com) (Quit: Connection closed for inactivity) |
| 2026-07-06 23:21:43 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) (Ping timeout: 265 seconds) |
| 2026-07-06 23:18:32 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) |
| 2026-07-06 23:16:49 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) merijn |
| 2026-07-06 23:09:51 +0000 | <ski> | iow, you can specify than you're repeating with an `m', with the special case `m = 0' meaning that you're not materially repeating |
| 2026-07-06 23:07:37 +0000 | <ski> | this is also why, imho, `mod n 0' ought to be equal to `n' (and `gcd 0 0 = 0' is also a bit related) |
| 2026-07-06 23:07:37 +0000 | GdeVolpiano | (~GdeVolpia@user/GdeVolpiano) (Ping timeout: 254 seconds) |
| 2026-07-06 23:06:58 +0000 | <ski> | going back to considering the ring of integers `|Z' / `Integer' as "cyclic"; well, in terms of congruence / modular arithmetic, `|Z' is basically the same thing as `|Z / (0 * |Z)' (integers modulo zero) (`a = b (mod m)' means `m | a - b', now, `0 | n' only holds for `n = 0' (`a | b' means `exists k : |Z. k * a = b'), so this amounts to plain equality, `a = b') |
| 2026-07-06 23:06:22 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) (Ping timeout: 276 seconds) |
| 2026-07-06 23:05:49 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) (Ping timeout: 265 seconds) |
| 2026-07-06 23:01:26 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) merijn |
| 2026-07-06 22:59:44 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) |
| 2026-07-06 22:59:21 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) (Remote host closed the connection) |
| 2026-07-06 22:58:11 +0000 | <ski> | (otoh, if you take e.g. the degree `m' of a nilpotent element `r' in a ring (e.g. a nilpotent (square) matrix), this is expressed by `forall n. m =< n <=> r^n = 0', where we use the additive ordering, because `r^(n+i) = r^n * r^i = 0 * r^i = 0'. so if it holds for some `n', it also holds for all greater integers in the usual ordering) |
| 2026-07-06 22:55:46 +0000 | <ski> | z^i = z^i' which is not in general `1', for natural numbers `i')) |
| 2026-07-06 22:55:40 +0000 | <ski> | (`m' is the smallest (in divisibility order) `n' such that the `n'th power of `z' is `1'). here, if `z^n = 1' holds for an `n', it also holds for all integers `c*n' greater than `n', in the *divisibility* (aka geometric or multiplicative) ordering, `z^(c*n) = (z^n)^c = 1^c = 1', but *not* simply for all integers greater than `n' in the usual (arithmetic, additive) ordering (because `z^(n+i) = z^n * z^i = 1 * |
| 2026-07-06 22:55:34 +0000 | <ski> | (the reason for using the divisibility ordering is that if `s' repeats after `n' steps, then it also repeats after `c*n' steps, for any integer `c', iow for all integers that are greater in the divisibility ordering. same thing happens for e.g. (complex) primitive roots of unity (and more generally the order of an element in a group), `z' is an `m'th primitive root of unity, if `forall n. m | n <=> z^n = 1' |
| 2026-07-06 22:54:07 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) |
| 2026-07-06 22:52:19 +0000 | ttybitnik | (~ttybitnik@user/wolper) (Ping timeout: 260 seconds) |
| 2026-07-06 22:50:43 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) (Ping timeout: 264 seconds) |
| 2026-07-06 22:48:29 +0000 | <EvanR> | division by zero for fun and profit |
| 2026-07-06 22:47:09 +0000 | <ski> | `s' has period `0' according to this definition if it doesn't have a positive period |
| 2026-07-06 22:47:03 +0000 | <ski> | now, it makes a certain sense to consider such infinite strands as "cycles of period zero", because if you consider a sequence `s : |Z -> A', it being periodic with period `p' is expressed by `forall n. p | n <=> forall k. s (k + n) = s k' (`p' is the least (in divisibility ordering) `n' such that `s' repeats after `n' steps), and if you pick `p = 0' (being the greatest element of the divisibility ordering), |
| 2026-07-06 22:46:40 +0000 | <ski> | EvanR : a permutation (a self-bijection `f : A -> A') consists of a number of disconnected cycles. if it is infinite, it may also contain some infinite orbit "strands" |
| 2026-07-06 22:46:04 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) merijn |
| 2026-07-06 22:41:40 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) (Ping timeout: 261 seconds) |
| 2026-07-06 22:35:23 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) |
| 2026-07-06 22:35:21 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) (Ping timeout: 256 seconds) |
| 2026-07-06 22:34:17 +0000 | target_i | (~target_i@user/target-i/x-6023099) (Quit: leaving) |
| 2026-07-06 22:30:45 +0000 | merijn | (~merijn@host-cl.cgnat-g.v4.dfn.nl) merijn |
| 2026-07-06 22:25:19 +0000 | traxex | (traxex@user/traxex) (Ping timeout: 245 seconds) |
| 2026-07-06 22:21:15 +0000 | eggplantade | (~eggplanta@2600:1702:8450:c370:68e2:8150:e673:9bfc) (Ping timeout: 243 seconds) |
| 2026-07-06 22:20:20 +0000 | CiaoSen | (~Jura@dynamic-077-181-231-255.77.181.pool.telefonica.de) (Ping timeout: 265 seconds) |
| 2026-07-06 22:19:33 +0000 | merijn | (~merijn@62.45.136.136) (Ping timeout: 255 seconds) |
| 2026-07-06 22:19:22 +0000 | CryptLab | (~NSA@145.79.196.162) (Ping timeout: 265 seconds) |
| 2026-07-06 22:19:06 +0000 | Fijxu_ | (~Fijxu@user/fijxu) fijxu |
| 2026-07-06 22:18:53 +0000 | Fijxu | (~Fijxu@user/fijxu) (Ping timeout: 265 seconds) |
| 2026-07-06 22:16:49 +0000 | Crypt_Lab | (~NSA@145.79.196.162) CommanderBond007 |