Mathematics - 2011-10-14 (page 2 of 2)

robjohn

13:01

@Gortaur: Here is the answer :-)

Gortaur

@robjohn: beautiful ) how did you guess about that function?

robjohn

I thought way too long about it though

J. M.

@robjohn Wait, isn't that how most of us solve problems anyway? :D

robjohn

No, it was suggested by the fact that when looking below 1, f is varying like g, whereas when looking above 1 I was always looking at exp(sum(x_n))>=prod(1+x_n)

Gortaur

+1

I still was not sure if it is right or not - that's why I made all small steps trying not to use inequalities too much

otherwise if on one step I cannot continue the proof, that would be a sign that counterexample can be developed from that point

robjohn

13:08

I tried to show that g_n <= f_n (fail) and g_n <= exp(f_n-1) (fail) and then it struck me to combine them

The first works for f<1 and the latter when f>1

Gortaur

I see

robjohn

phi(f_n)-g_n is non-decreasing function of n

Gortaur

no problem with your fix

I understand then increasing almost every time means non-decreasing

robjohn

Too much thought for a problem with so little scope. I don't know if I can use the result anywhere else. Maybe if I were an economist.

Gortaur

I agree that it was too much for me too, but I like the concept that f bounds g if it is non-profitable

that means that whenever you're in [1,2) do not invest the whole amount

robjohn

13:16

I was definitely intrigued by the concept, too. I drove out to visit our tax preparer yesterday (a couple of hours total) and thought about it during the whole drive.

I will have to see of our financial consultant has ever seen this result.

@Gortaur: now I can read yours :-)

Gortaur

a?

@robjohn you were thinking about it during the whole drive - and you tell me the story that you were about to crash some time ago )

I'm trying to forget about math when driving

robjohn

crash? are you referring to the fender bender when the brakes went out 35 years ago?

Gortaur

ok, maybe I mistranslated, but

I'm working in safety analysis of systems a bit and just considered the worst case scenario )

robjohn

as long as I don't use my whiteboard too much, math doesn't interfere with my driving.

Gortaur

that was the second thing I wish to danger you about

but I'm happy that you don't use it too much

robjohn

13:27

:-)

Gortaur

)

I'm so tired and my thoughts are so slooow

J. M.

...er, it's counterproductive to be doing mathematics when exhausted. Get yourself rested.

Gortaur

I'm not doing it for the last 30 minutes

robjohn

Ah, time to wake my wife and walk the dog. @Gortaur, I will finish reading your answer when I return.

Gortaur

@JM taking a rest with nfs hp. requires no thoughts at all

@robjohn: thanks, see ya

hope to go play a piano a bit, but there is a meeting in that room (

Asaf Karagila

13:50

Which LaTeX editor do you guys use?

Gortaur

win edt+sumatra pdf

@AsafKaragila: and you?

J. M.

I haven't written a paper with LaTeX... yet.

Asaf Karagila

Texmaker, it has both linux and windows versions. So I can use it both in my office and at home.

QED

hi

Gortaur

14:10

@QED: hi

Asaf Karagila

Ho

QED

I was thinking about nonstandard models of arithmetic

is anyone interested in that

Asaf Karagila

Not personally, it does sound somewhat interesting though.

@QED: You have to remember something about nonstandard models, though. They don't know that they are nonstandard.

Well, in some cases anyway.

QED

what do you mean?

how could it know?

Asaf Karagila

14:26

In a nonstandard model of PA, let omega be the least infinite integer. What is the predecessor of omega?

Hm. It comes to me that I'm not familiar enough with nonstandard models of PA.

QED

It is not the case that omega = S omega, since you can disprove that by induction

Asaf Karagila

I remember to recall reading some answer in which they said that nonstandard models of PA are ultraproducts of Z

QED

of course ultraproducts come up in the proofs of theorems used to exhibit the models

I haven't seen such an explicit use of ultraproducts as in Robinsons analysis

no. I was thinking of ultrafilters

Asaf Karagila

I was thinking of this question, I think: math.stackexchange.com/questions/37418/…

This search on MO has more information: mathoverflow.net/search?q=%22non-standard%22+ultraproduct

QED

interesting post

robjohn

15:03

@AsafKaragila I use BBEdit and CMacTeX

rather bare bones for editing. :-)

t.b.

@Asaf: Aquamacs + tetex (or whatever else it is called now) + skim (pdf viewer)

Jacopo Notarstefano

@AsafKaragila TextMate

robjohn

@Gortaur: I edited your answer a bit. I have noticed that f_{\sigma_{k+1}} is hard to distinguish from f_{\sigma_k+1}

I am still reading it, but that caused me to pause a bit.

J. M.

I think you need to start considering alternative notation when your subscripts have subscripts... :)

robjohn

It's needed here since we are considering subsequences of sequences.

t.b.

15:14

@JM Did you now manage to convolve your generating functions?

robjohn

I guess he could call f_{\sigma_k} h_k

QED

f_i,j

J. M.

@t.b. Oh yeah. :) It wasn't for an m.SE question though; it was something that popped up in a problem a friend was asking me for some help with...

Gortaur

@robjohn: many thanks. is it readable at least? I guess it's hard to go through notation there

t.b.

@JM very good :)

robjohn

15:16

It is readable, but I have to keep reminding myself where the sigmas and taus are

t.b.

okay guys, see you around, I gotta go and catch some sun

Gortaur

hm, that means that my explanation is not clear - I thought that should the most clear part what sigmas taus do dnote

Gortaur

16:07

Ok, folks I'm leaving soon - have a nice rest of Friday and cool weekend

robjohn

16:54

@Gortaur: I've commented a bit on your answer. I am still working on the last bit.

Oops, he's gone.

J. M.

He'll see it when he gets back... assuming he reads transcripts.

robjohn

Sometimes when I get back the transcript is huge.

I often don't get to it all.

J. M.

17:20

Also, since you @-lerted him, it should show up as a red indicator in the upper left corner once he gets to the main site...

Oh my, a question by JDH again? I expect a rain of upvotes again... :D

Gortaur

17:59

@robjohn: thank you for the comments, I will take a look if it can be fixed

robjohn

Ack. I hate it when I answer a question and then realize it was a homework question. I just deleted my answer. Blah.

J. M.

@rob: maybe you should consider adding [homework] to your list of ignored tags... :)

robjohn

Heh. I don't mind giving hints; I just hate not noticing the tag.

@Gortaur: I still have something on which I might also comment.

J. M.

On the other hand, that Bohr thing is quite old, so...

Gortaur

@robjohn: oh no... give me mercy ))

I should promise to myself never post answers with which I feel uncomfortable

robjohn

18:04

Did I say too much?

Gortaur

no

that was a kind of joke

show me no mercy

robjohn

I think that it might be fixable.

I was trying things like that before I got sideswiped by the answer, but I wasn't able to make them work all the way.

That doesn't mean that they might not work.

Gortaur

I hope they will - but I don't have an opporunity to fix'em now

that's a pity, hate leaving confusing answers

I already lost reputation in eyes of some members

J. M.

Hmm, new Ubuntu...

I was already wondering what they'd do for the letter "O"...

Gortaur

18:20

@JM which version?

J. M.

"Oneiric Ocelot" is what they're calling the new one...

Gortaur

i prefer kubuntu, though I didn't have a change to switch to it

J. M.

Actually, I'm on Xubuntu. :) I quite like Xfce more than GNOME...

Clinton Freeman

Had a quick question that probably doesn't warrant a full post if anyone knows

J. M.

Fire away.

Clinton Freeman

18:24

if i have a normal vector to a plane and cross product it with cardinal axis that corresponds to the component of least magnitude, will that give me some randomly oriented vector that lies inside of the plane?

J. M.

...where is the randomness coming from?

Clinton Freeman

well

i mean, "random"

Gortaur

you mean 'some vector', right?

Clinton Freeman

right

J. M.

The plane is "random"?

Gortaur

18:25

I don't know what is the cardinal axis

@JM )) that's not about probabilty

Clinton Freeman

x y or z

(1, 0, 0) (0, 1, 0) or (0, 0, 1)

Gortaur

yes

J. M.

@Gortaur: nono :) I mean if he had some randomly constructed plane from, say, three randomly chosen points...

@Clinton: oh, the coordinate planes...

Gortaur

@ClintonFreeman V x U always normal to V

no matter what is U

so if V is normal to the plane P

then for any U you have V x U = [V,U] \in P

if your vectors are free, of course

@JM I was thinking that 3 randomly taken points almost always define the plane uniquely and then had a question why there is no a shorthand a.a. for almost always

J. M.

@Gortaur: I don't know either. :)

I've forgotten the details, but the probability of collinearity I remember to be quite low...

Gortaur

18:32

but not less than zero (!)

Clinton Freeman

okay cool

thanks!

Gortaur

you're welcome, be careful with vectors

they're sharp

J. M.

@Gortaur Certainly. :P

Jacopo Notarstefano

Ah! Apparently, "autonomous sets" are more commonly called "modules"

And "module decomposition" is a well-studied topic.

This makes my last question somewhat useless...

Gortaur

does anybody know ergodicity here?

robjohn

18:37

@ClintonFreeman If the coordinate of least magnitude has component 0, the cross product will be 0, which technically is in the plane, but is it useful?

Clinton Freeman

hmm

Gortaur

@robjohn: a, you've fixed it

there was a 2nd column

robjohn

@Gortaur: is it a big city?

Jacopo Notarstefano

/me facepalms

Gortaur

@robjohn Moscow is quite big, yes

robjohn

18:41

@Gortaur It was a bad joke about ergodicity. I think I have heard of Moscow, though.

Gortaur

note that Moscow in Idaho is quite small though

Clinton Freeman

if the particular component were zero then i could just use the axis itself as the perpendicular?

Gortaur

@robjohn I didn't get it from the first time I read your post, so since Moscow is a default city, I know what to reply

robjohn

Moscow, IA
Zip code(s): 52760
Moscow, ID (city, FIPS 54550)
Location: 46.72977 N, 116.99684 W
Population (1990): 18519 (6748 housing units)
Area: 12.4 sq km (land), 0.0 sq km (water)
Zip code(s): 83843
Moscow, KS (city, FIPS 48600)
Location: 37.32547 N, 101.20634 W
Population (1990): 252 (105 housing units)
Area: 0.5 sq km (land), 0.0 sq km (water)
Zip code(s): 67952
Moscow, OH (village, FIPS 52416)
Location: 38.86035 N, 84.22849 W
Population (1990): 279 (109 housing units)
Area: 1.0 sq km (land), 0.0 sq km (water)

A few Moscows

old censuses

Sorry if that spammed people.

Gortaur

in Kansas people use to live two in the house

J. M.

18:45

If you had typed those out, I'd have been more impressed... :D

Gortaur

while in Idaho there are usually three inhabitants in one house

@ClintonFreeman you mean, as a vector in the normal plain? yes

Clinton Freeman

yah

J. M.

But Gortaur, I think we're not in Kansas anymore... :)

Gortaur

@JM why I should remember it?

J. M.

Oh, sorry. I had this in mind...

Gortaur

18:50

is it a phrase from that movie?

robjohn

@J.M. but Kansas is on the plains

J. M.

Well, there wasn't anybody there named "Gortaur", so not exactly... ;D

robjohn

Toto, I don't think we're in Kansas anymore.

If it's not in the movie, it is definitely attributed to the movie.

J. M.

Well, I can't upgrade with the browser open. and I'm sleepy anyway. See you guys later.

robjohn

@J.M. off to work?

Have a good day, see you later.

Jacopo Notarstefano