Top.Mail.Ru
? ?
Concepts and Contracepts [entries|archive|friends|userinfo]
the buhrger

[ barely used website | Why Look Here? ]
[ boring data | livejournal userinfo ]
[ the past | journal archive ]

Inner Factor Ratios for measuring roundness [Tue 2013/07/09|17:17:00]
the buhrger
[Tags|]

almost immediately after posting about using gppf as a measure of roundness, another one of course crossed my mind. i've been calling it IFR, for Inner Factor Ratio. by "inner factor"(s), i mean the two factors closest to the number's square root. so, for instance, the factors of 12 are 1, 2, 3, 4, 6, 12. the two numbers closest to its square root are 3 and 4, so its IFR is 4/3.

i came to this measure while working on the gppf stuff, actually. making lists of factors of a number, i noticed that some numbers had lists with really large gaps in the middle (e.g. 57: 1, 3, 19, 57, the gap being between 3 and 19), and some didn't (e.g. 56: 1, 2, 4, 7, 8, 14, 28, 56, the gap being between 7 and 8), and that the round numbers tended to be the ones with small gaps. (remember that, as we're working multiplicatively, the "middle" of a list like this is necessarily the square root of the number.) of course (again with the multiplicative), it makes more sense to measure the size of this gap using a ratio instead of a difference. thus, Inner Factor Ratio.

an obvious concern is whether the IFR is well-defined for squares. if you consider something like 36: 1, 2, 3, 4, 6, 9, 12, 18, 36, the middle/square root is 6, which is actually in our list. so do we want the ratio of 6 and 4, or the ratio of 9 and 6? conveniently, these two ratios will always be the same. ( proofCollapse ). so the IFR is even well-defined in this situation.

by definition, the IFR is always greater than 1 (it being the ratio of two numbers, with the bigger one on top), and less than or equal to n (which upper bound is attained if n is a prime number). one could thus normalize the IFR by dividing by n, or by taking the logn, but i'm disinclined to do this. in at least some cases, the IFR has some nice scaling properties. IFR(k2n) ≤ IFR(n) because if a, b, are the inner factors of n, then ak, bk are candidates for the inner factors of k2n. we also note that if p is prime, the factors of pn are all the intermediate powers of p, and thus the IFR of pn for any n is necessarily p. so in some sense the IFR is scale-insensitive.

based on those results, it's tempting to make what i call the strong domination conjecture: IFR(mn) ≤ min(IFR(m),IFR(n)). this is false. consider, e.g., 606 = 6 * 101. IFR(6) = 3/2 = 1.5. but IFR(606) = 101/6 = 16.8333 is significantly larger than 1.5. a nerfed version, however, the weak domination conjecture IFR(m*n) ≤ max(IFR(m),IFR(n)) is true. ( proofCollapse )

the asymptotics of IFR are still somewhat vague. i've tabulated it up to 1753 so far. mean values will be dominated by primes and biprimes, so i've looked at medians instead. the median value of IFR(n) seems to stay fairly consistently around n0.26, but, as i say, that's still somewhat vague. minimal values are attained by numbers in the form n(n+1) (A002378). somewhat more problematic, lurking close to these are some biprimes, specifically those that are products of twin primes. my æsthetic has a problem calling a biprime "round", even one with a nice IFR like 41 × 43 = 1763.

of course, i'm posting about this now because just this morning, thinking about this biprime issue lead me to yet another way of measuring roundness, which i like better! but i'm gonna play with it a bit before writing it up.
linkwhat?

still round after all these years [Thu 2013/01/17|20:56:00]
the buhrger
[Tags|]
[for some bizzare reason i'm telling you how i feel |i'm just a geek]

because i never really stop thinking about anything, i'm going to again go on for a little bit about quantifying the roundness of numbers. way back here, i mused about looking at the greatest prime factor gpf(n) of an integer, and defining an integer's roundness as logngpf(n). at the time, i commented:
the problem with this method is that it tends to overprivilege prime powers. any number of the form n = pk for p prime will end up measuring as very round in this calculation. in fact, this calculation establishes them as the archetypal round numbers. which feels a little off to me: a round number should play nicely with more than one prime.
for some reason, it took three and a half years to see the obvious way around that. (which raises the obvious question: what do i mean by "obvious"?) instead of looking at the greatest prime factor of n, look at the greatest prime power factor of n. gppf(n) has some curious properties that gpf(n) doesn't. it's an ultimately increasing function, for instance: for any n, there's a threashold Nn such that m > Nn means gppf(m) > n. isn't that nifty?
i've been tabulating values for a few days now (thanks to A034699 for getting me started), and still don't have a really good handle on its asymptotic behaviour. but it looks like a fascinating approach. notably round numbers (i.e. ones that have a lower roundness index than anything before) so far are 6, 12, 20, 30 (the first one with gppf(n) less than the square root of n), 60, 210, 420 (the highest number with a gppf of 7), 840 (the highest number with a gppf of 8), 1260.
i've only tabulated to 1763 so far (i'm doing it partly by hand, using a moderately buggy smidge of common lisp code gakked from here), but the next time i get the proper alignment of free time and will, will likely throw maple at the problem. i'm expecting good things from 2520 and 27720, for "obvious" reasons.
linkwhat?

when is radian day? [Wed 2011/06/29|17:02:00]
the buhrger
[Tags|]

both π day and the more recent τ day (both more recent = it was yesterday and more recent = it hasn't been celebrated for as many years) suffer from the same problem: they depend on the decimal expansion of one or t'other of the circle constants. using base 10 is precisely the sort of arbitrary choice that going to radian measure of angles is attempting to avoid. thus, one wonders, is there a more appropriate way to select a day to celebrate all things circular?

one obvious choice would be to celebrate when the earth has moved through one radian of its orbit, i.e., when the distance it's travelled (from, e.g., perihelion) is equal to its radius from the sun. there are far fewer mathematically arbitrary choices involved in this. perihelion is around january 4. and 365.242199/τ (365.242199/(2π) for the π-ous) is 58.130101389 = 58 days + 3 hours + 7 minutes + 20.76 seconds. doing the math suggests that radian day would be around march 4th, which conveniently is also GM's day.

two problems:
  1. someone else thought of it first. technically, this is the opposite of a problem. although, by starting the radian-clock rom perihelion (i suppose one also might want to use winter solstice?), my radian-day ends up on a slightly different date.

  2. this one, however, is a problem: the earth's orbit famously ain't a circle. thank you, johannes kepler! it's an ellipse. and calculating lengths along an ellipse will require futzing about with, well, elliptic integrals (guess why they're called that). maybe later.
linkwhat?

who are you, and what do you want? [Tue 2011/04/19|10:30:00]
the buhrger
[Tags|]

jugglingstore, mian1wzd, neverexost, and renegadesufi, i'm curious about how you came to friend me on LJ. where did you find my journal? you realize that, as per my friending policy, discussed on my profile page, i'm not gonna be friending you back any time soon. right?

just wondering. this prolly reads snarkier than it should. you're of course welcome to hang out. just don't expect much of interest, except for math. pretty much everything else is locked down.
linkjust 1|what?

my first filk? [Sat 2011/03/05|18:12:00]
the buhrger
[Tags|, ]
[for some bizzare reason i'm telling you how i feel |geekygeeky]
[i've got a brain like a jukebox |"dance yrself clean" - LCD soundsystem]

ever since first seeing walken dance his way through the video for Fat Boy Slim's Weapon of Choice it's been obvious to me that the song needed a mathematical filking. thus, i present

Axiom of Choice

Don't be shocked by Cohen's and Gödel's voice
Check out my new axiom, axiom of choice
Don't be shocked by Cohen's and Gödel's voice
Check out my new axiom, axiom of choice, yeah
Listen to Cohen's and Gödel's voice (aah...)
You can check it on out, it's the axiom of choice, yeah

You well-order this this
Or you well-order that
You well-order this
Or you well-order that
Or you well-order this
Or you well-order that
Or you well-order us

You well-order this
Or you well-order that
You well-order this
Or you well-order that
You well-order...

Principle of Hausdorf, a lemma due to Zorn
Principle of Hausdorf, and a lemma due to Zorn
Principle of Hausdorf, and a lemma due to Zorn
With a Principle of Hausdorf, uh, we're gettin' warm

Ambiguously known?
As constructivists, they moan
At pure existence proofs
and my pal
I guess you just don't realize
It's gone beyond doublin' size
We're mathematical sleuths
For new truths

some parts of this are clearly stronger than others. suggestions for improvement are very much welcome.

i note also that the original material has several other lyrics, on loops, i think not performed by the main vocalist. ( under the cut, some suggestions on appropriate rewritingsCollapse )
link4 already|what?

the danger of thinking and walking at the same time [Tue 2010/09/28|20:22:00]
the buhrger
[Tags|]

on a walk the other day, i noticed that several of the passing (three-digit) license plate numbers i factored ended up having a greatest prime factor (gpf) of 31. as in, i think, four in six consecutive plates. of course, i wondered if there was anything to that. as the largest three-digit number is 999 (roughly 1000), and the square root of 1000 is roughly 31, i immediately conjectured that the average greatest prime factor around the number n should be roughly the square root of n. stated slightly more formally: if one defines a function f of x to be the sum of gpf(n) for all n between 1 and x (what is the gpf of 1? 0? 1?), then either f(n)/n or f′(n) is roughly the square root of n. equivalently, i conjectured that f(n) is O(n1.5)

conveniently, the OEIS page for the gpf sequence links to the first 100000 gpf values. slapping those in a handy dandy spreadsheet and doing the relevant calculations suggests that (at least as far as i've run the calculations so far) f is more like O(n1.76) or even higher. i'm starting to suspect that it might be O(n?log n) for some value of ?, but haven't bothered playing around with that yet.

for all i know, someone's already figured this out. one of these years, i'll have to figure out how to search the literature.

(oh, also? the most common gpf for numbers between 1 and 999 is 7, nowhere near 31. yesterday's walk-observation was almost certainly just ramsey theory messing with me.)
linkwhat?

kolmogorov complexity of integers [Tue 2010/08/24|08:51:00]
the buhrger
[Tags|]

( cut for do you really care?Collapse )
linkwhat?

the question of the morning [Sun 2010/06/27|06:56:00]
the buhrger
[Tags|]

why is it that the african nation called the gambia is called "the gambia" instead of just "gambia"? i recognize that articles on country names are common in other languages. auf franzözisch, zum beispiel, man sagt "le canada", aber auf englisch man sagt nicht "the canada". i remember that we used to call ukraine "the ukraine", but that stopped somewhere along the way. so any guesses on the gambian (or should i say "the the gambian") situation?
link6 already|what?

the normalized totient, or, who needs repeated prime factors anyway? [Mon 2010/05/24|09:23:00]
the buhrger
[Tags|]

i continue to think about roundness quantification when i've got nothing better to do. another approach, one might even say a classic approach, would be to ask how many numbers smaller than n share factors with n. if there are lots of them, n is round. not so many, not so round.
quantifying this is pretty easy, thanks to Euler's totient function φ(n). you'll recall that φ(n) is the number of numbers smaller than n which are relatively prime to n (i.e. they have no common factors with n other than 1). so if φ(n) is small, n is round; if φ(n) is big, n is not. except that "small" and "big" are relative to the size of n itself, so we're better off normalizing it: φ(n)/n.
of course, there's a formula for φ(n) based on the prime factorization of n. if n=Πpiki, then φ(n) = n*Π(1-1/pi) (products taken over all prime factors of n). normalizing this gives φ(n)/n = Π(1-1/pi). easy peasy.
one reason i'd previously dismissed this measure is that it fails to reward repeated prime factors. 6 and 12 and 18 and 24 and 36 are all divisible by 2 and 3, and thus all end up with a normalized totient of (1 - ½)(1 - ⅔) = ⅓ — nice and low, to be sure, but all give the same result. but i'm starting to warm to that idea. is 20 really much rounder than 10?

in any case, in this measure, maximal values are given by primes (and, of course, given the above, prime powers), with normalized totient values of (p-1)/p, tending to 1. minimal values are given by primorials, and i believe (but haven't yet sorted out a proof, so i might be worng) their normalized totients will tend to zero.
linkwhat?

canadian cylons [Fri 2010/03/26|21:01:00]
the buhrger
[Tags|, ]

( silly but not cut for that. cut for statistics and mild-to-moderate BSG spoilers.Collapse )
link4 already|what?

dealing with rural overrepresentation|urban underrepresentation [Sat 2010/02/27|08:10:00]
the buhrger
[Tags|, ]

when my feeds barfed up daveberta's recent post on the proposed electoral boundaries for the next provincial election, i mostly just glanced at the map of edmonton, noted that the "mill woods" riding will be closer to actually being mill woods than it is now, and let the topic pass from my mind.
then, yesterday evening, my del.icio.us feeds barfed up (i'm using "barfed" in a non-pejorative sense, here, in case that's not clear. it just seems like a fun word to use this morning) an article from the local paper which talks about the perpetuated inequalities of the proposed new districts. which won't be a surprise to anyone who's followed alberta politics at any time in the last few decades. as has been the case for a long time now, rurual areas are overrepresented and urban areas are underrepresented. and the tories always take nearly all the rural ridings, so it's in their interest to keep things that way. all of this is understandable, if not fun.
but again the argument gets trotted out that one can't make rural ridings too large, or they become unweildy for an MLA to represent. which is likely true.

so on the one hand, rural ridings have to be kept small enough that they're representable, and on the other hand, urban ridings get underrepresented.

the solution to this is, of course, obvious.

why does an MLA representing a rural riding of 24,900 people have the same voting power as an MLA representing an urban riding of 47,636 people? (those are the ones cited in the newspaper article) why not weight their votes? in the case of these two ridings, the urban MLA's vote should count for nearly double that of the rural MLA. they still both represent their populations, but as the urban chappie is representing more people, the urban chappie gets more vote. if that sounds complicated, well, i think there are devices around today that are good at handling numbers: they're called computers spreadsheets pocket calculators, and you can get one for under $20 that would handle all the calculations required for this to be implemented. if that's too much for the provincial legislature to cough up, i'll happily donate one.

if this sounds just slightly kinda familiar to you, it was pondering this a couple of years ago that lead me to a much more radical proposal, which is the logical conclusion of this sort of thinking. of course, that system ended up changing so many things that it was hard to know whether it would work. in the case of this one, on the other hand, it's a small change.

so, that's the speculative policy for the morning. would it work?
link8 already|what?

just asking [Fri 2010/02/12|19:12:00]
the buhrger
[Tags|]

am i the only person who's sad to find that google only gives two hits for the phrase "commuter catapult"?
link2 already|what?

coming down is the hardest thing? [Tue 2009/08/25|08:43:00]
the buhrger
[Tags|, ]

i remember around the time that i was learning to drive (and just before) that i wondered started wondering how the process of driving affects the way the mind works. i'm sure one place it shows up is in habits of attention: check mirrors, check speedometer, keep eye on road, watch for moonwalking bears, etc. i'm sure there are habits of perception as well. i wonder what else.
linkwhat?

mountain driving assist? [Tue 2009/08/11|09:22:00]
the buhrger
[Tags|, ]

so here's the kooky idea i had recently about mountainous driving: you know how mountain roads inevitably include the occasional rather steep section? and the cars driving up have to work extra hard to get up, and the cars driving down have to sit on their brakes so that they don't lose control of their vehicles. (i'm exluding teenage "look how fast i can go" joyriders from the equation, i suppose.)
so, phrased another way, the folks going down have this excess kinetic energy that they'd rather be rid of. and the folks going up could make use of some extra kinetic energy.
so the idea, vaguely inspired by railguns, would be to use some sort of magnetic doohickey to suck some of the energy|momentum|velocity off of the downward traffic (this half also looks a little like a hybrid car's regenerative braking), and pump the resulting electricity across the median to power the magnetic doohickey for the upward traffic.

i'm sure there are seventeen reasons why this is a bad idea.
link23 already|what?

random question #316523 [Mon 2009/07/27|19:59:00]
the buhrger
[Tags|]

why is it that, with electronic car door locks becoming so ubiquitous, electronic house door locks are still comparatively rare?
link34 already|what?

cover your eyes! it's a little more math! [Sat 2009/06/20|21:33:00]
the buhrger
[Tags|]

very preliminary investigations suggest that the sum of the roundness indices for three consecutive numbers may always be at least 1. i'm gonna have to sic some more intensive calculatory on the issue. if this is so, it may have mild implications for factoring algorithms. (an obvious hard upper bound for this sum is three, which assuming something that seems obvious to me about twin primes is true, can be improved to at least 3 — logn9, possibly further.)

EDIT/DISPROOF: the sum of the roundness indices for 1078, 1079, 1080 is 0.99555082. it likely helps that the index for 1080 is so ridiculously low. revised conjecture: there may be no positive lower bound (i.e. the lower bound may be zero, although if so, it's gonna take its own sweet time getting there).
linkwhat?

roundness quotient and roundness index [Sat 2009/05/30|20:11:00]
the buhrger
[Tags|]

( cut for gratuitous mathematicsCollapse )
link4 already|what?

tie-dye for geeks [Sun 2009/05/17|20:02:00]
the buhrger
[Tags|]

( cut for moderately heavy mathematicsCollapse )

if you've been skipping the math, you can hop in here and look at the pix:

( cut for pretty pixturesCollapse )
link2 already|what?

things that are full of awesome [Thu 2009/02/12|16:18:00]
the buhrger
[Tags|]

folk who aren't fans of starwars will be less impressed by these.



two sleds:


yes, those are sleds. want to see them in action? there's vid here with those and many others.

thanks to ironymaiden and wil, respectively for choice linky.
linkwhat?

i've said it before, and i'll say it again [Tue 2008/12/30|10:43:00]
the buhrger
[Tags|]

simple minds, simple pleasures; complicated minds, simple pleasures.

( ceci n'est pas un cut-tag )
link6 already|what?

going all the way? [Sat 2008/11/29|10:06:00]
the buhrger
[Tags|, ]

as the cowboy for social responsibility notes, there are three ways that the canadian taxpayer has been supporting political parties:
  1. Rebates to parties and candidates for campaign expenditures;
  2. Tax credits for donations to parties and candidates; and, most recently,
  3. Core support for political parties based on the number of votes they received in the last general election.
if the currently still governing conservatives are really interested in no longer forcing the taxpayer to support politicians, why not cut off all three?
link4 already|what?

my second-favourite percussion instrument [Tue 2008/11/11|09:40:00]
the buhrger
[Tags|, ]
[i've got a brain like a jukebox |all cuíca all the time]

recentlich i mentioned the cuíca to someone, and they hadn't heard of it. so this playlist is all about my second-favourite percussion instrument, the crazy monkey drum!

traditionally, the instrument is used in samba bands. my familiarity with brasilian samba is woefully minimal. anyone want to recommend some good starting points?
link7 already|what?

some longer reads/watches [Sun 2008/11/09|09:49:00]
the buhrger
[Tags|, , , , ]

partly inspired by anathem's advocacy of activities that require an attention span, this weekend has seen me spending a fair amount of time reading/watching lengthier stuff on the 'puter. if you've got the time, and would like to join in, here's what i've been perusing:the irony of all of the above being, of course, that i've not been reading as much of anathem as i might like. but there's an obvious solution for that.
link2 already|what?

a job for roäc, son of carc? [Tue 2008/11/04|18:57:00]
the buhrger
[Tags|, ]

there's been some kvetchery recently about local litter caused by smokers. it brings to mind a video i came across not too long ago on TED. i wonder if we could bribe train the local corvids to pick up butts in exchange for peanuts?
link4 already|what?

a public service announcement [Thu 2008/10/30|07:29:00]
the buhrger
[Tags|, ]



for the person driving the ford pickup with license plate FUN 659 and everyone else, a reminder from alberta's Use of Highway and Rules of the Road Regulation:
93(1) At a place where there is a crosswalk, a pedestrian has, unless otherwise directed by a peace officer or a traffic control device, the right of way over vehicles for the purpose of crossing the roadway within the crosswalk.
believe it or not, this applies to you too!
link9 already|what?

flaky LJ? [Mon 2008/10/27|09:20:00]
the buhrger
[Tags|]

two days in a row, around 8am, LJ has temporarily cut out for me. is anyone else experiencing the same thing?
link5 already|what?

the main spring turning the world around [Tue 2008/09/23|17:58:00]
the buhrger
[Tags|, ]

so if corporations get to be treated as persons people before the law, why aren't they treated as people before the tax code? why are there different rates for corporate taxes than there are for income taxes? heck, if each human gets one vote, why not give each corporation one vote too? EDIT: and also let them be sent to jail (what would that mean?) if they're convicted of criminal offences.

i have no idea if i'm serious or not.
link5 already|what?

my question for the candidates [Sun 2008/09/07|14:41:00]
the buhrger
[Tags|]

so, dropped writ. october 14th we're again allowed out of our cages to act democratic for a minute or two. hmn.

do all-candidates fora happen any more? i have a question i'd like to put to my local candidates. i have a sneaking suspicion i won't get the chance to ask all of them at once. i may have to e-mail them.

the Preämble:
Mr.Harper has described the current parliament as "increasingly dysfunctional". I tend to agree with this assesment. A minority parliament only works so long as the members of the various parties are willing to compromise and coöperate. There's been some of that in this last parliament, but much of the time the parties haven't been playing nicely together. Mr.Harper also expects "that we will have another minority". Again, i tend to agree wtih this assesment. If there is another minority, it may well set the stage for the same dysfunction to happen all over again.
Personally, i'd rather not have that.

the Question:
If elected, what would you do to make a minority parliament work?
If there is another minority, what would your party do differently to make it work?

the Caveät:
If your answer involves blaming other parties or people from other parties for the mess, you've just lost my vote.

any suggestions for wording modifications? the preämble seems kinda long... (and perhaps i should get rid of the diäreses?
link13 already|what?

pick a number - any number [Thu 2008/04/17|09:57:00]
the buhrger
[Tags|]

Poll #1172683 pick a number - any number
Open to: All, detailed results viewable to: All. Participants: 17

pick a number - any number

View Answers
Mean: 11.41 Median: 10 Std. Dev 5.45
0(0.0%)
1
0(0.0%)
2
0(0.0%)
3
1(5.9%)
4
0(0.0%)
5
3(17.6%)
6
0(0.0%)
7
2(11.8%)
8
1(5.9%)
9
1(5.9%)
10
0(0.0%)
11
0(0.0%)
12
2(11.8%)
13
0(0.0%)
14
0(0.0%)
15
1(5.9%)
16
0(0.0%)
17
4(23.5%)
18
0(0.0%)
19
2(11.8%)
20
0(0.0%)
link

directly proportionally representative democracy? [Mon 2008/03/10|19:34:00]
the buhrger
[Tags|, ]

so here's some more hare-brained speculative policy. as always, i'm not saying this is a good idea. or a bad idea. just that it's an idea, and i wonder:
what would happen if every vote counted?
even proportional systems eventually undercount or overcount some votes. here's a system that wouldn't do that:

suppose in the electoral district of the Borough of Shmeng, there are three candidates running for elected office, Alice, Bob, and Charlie. when the votes are tallied, the numbers look something like this:
Candidatevotes received
Alice2342
Bob1729
Charlie606

under the system i'm conjecturing, the strange thing that happens is that all three of them get to vote in the legislature. but they don't have equal votes. Alice's vote counts 2342, Bob's counts 1729, and Charlie's counts 606. so Alice's voice is louder than either of her competitors, but on issues where Bob and Charlie agree, they have more clout than Alice does.

there are some obvious logistical problems with this, of course. is there enough room to seat all the candidates in the legislature? is there enough time for everyone to participate in debate? how much do the elected officials get paid? (i'd be inclined to prorate remuneration based on votes received.) there may be others. feel free to identify them.

aside from proportional effects, i'm curious as to what a system like this would do to politics over time. suppose that a voter, call her Drew, was one of the 606 voters for Charlie in the above election, and decided that Charlie really hadn't done a particularly good job of representing her point of view. in the next election, Drew decides to do something different. Drew votes for herself. Drew doesn't formally campaign, just registers as a candidate and casts a ballot for herself. the next election, more people do the same. i'm sure most people wouldn't want to pay attention to all matters under discussion, and would rather let someone else do it - the process certainly allows for it - but i'm curious as to how many people would exercise this option, were it available.

other thoughts? criticism? speculations? does merely raising it overly muddy the waters of the PR movement? could it work? wujja think?
link26 already|what?

navigation
[ viewing | most recent entries ]
[ go | earlier ]