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Messages - ZetaX

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16
General Discussion / Re: Mathematics Help Thread
« on: March 31, 2015, 01:41:50 pm »
Quote from: Ispil on March 30, 2015, 08:33:29 am
So, here's some math that should be straightforward- polynomials.

Say I have the polynomial product (x+y)(x+z)(y+z)(x+y+z)=0. Now, I know that if you divide the entire thing by xyz, you are left with (x+y)2/z + (x+z)2/y + (y+z)2/x + 4x + 4y + 4z = 0.

My question is: other than fully expanding the entire thing out, is there any other way to get from one equation to the other?
In the following I am assuming that you want to show that those two equations are equivalent. You have several options:

a) Just expand. The long, tedious and boring way, but requires no thought at all. Best done when sleepy or if a slave is available.

b) You already know a neat factorisation of one equation. The second one has the same degree after multiplaying with xyz (you can see that without actually multiplying). Thus for them to be the same equation one only needs to check that the second one vanishes if x+y=0, y+z=0, z+x=0 or x+y+z=0. As the equations are symmetric (i.e. interchanging the variables does  not change either of them), we only would need to check vanishing if z+x=0 and x+y+z=0. So substitute z=-x and check for the second equation to be satisfied; then do the same with z=-x-y.
Still somewhat lengthy, but already shorter; has the disadvantage of needing the first one to be neatly factored which won't always be so.

c) Use the weak version the combinatorial nullstellensatz (Theorem 1.2 in http://www.tau.ac.il/~nogaa/PDFS/null2.pdf). In other words: polynomials are equal if they give the same result for enough choices of values for x,y,z.
In this case: multiply the second equation by xyz to get a polynomial. Then check if both polynomials from the two equations give the same values when plugging in sufficiently many values. Sufficiently many are e.g. each of the 5³=125 possible combinations of putting each of x,y,z to one of -2,-1,0,1,2. You would (by fully invoking that theorem and symmetry) actually only need to check 5+8+12 = 25 triples:
y=z=0, x from -2,-1,0,1,2
z=0, y from 0,1, x from -1,0,1,2
z from 0,1, y from 0,1, x from -1,0,1
A weird way to do it. Some slave or too much spare time recommended, but actually not that much work (those 0's cancel a lot).

17
General Discussion / Re: Mathematics Help Thread
« on: March 12, 2015, 10:19:34 am »
Quote from: Helgoland on March 10, 2015, 11:22:14 pm
Now if only my Topology Intro or Topology 1 had treated at least one third of the concepts in there...
I would say that a lot of it is more akin to homological algebra, which a lot of topology courses seem to skip over unless they cannot longer avoid it (e.g. Hatcher's textbook does this all the time). For beginner's courses this is maybe fine, but might still come back to bite you later.


Quote from: GreatJustice on March 10, 2015, 09:31:46 pm
First, if I have a prime and two integers x and y s.t p|x^2 + y^2 but p doesn't divide x or y, how would I go about proving that there is a k s.t k^2 is congruent to -1 (mod p)? I feel like I'm missing something quite obvious.
Essentially: just take k = x/y. As an element of a finite field this is directly defined; more classically (but actually the same), you would use xy^(-1), where z=y^(-1) is any integer inverse to y mod p, i.e. yz = 1 mod p.

Quote from: GreatJustice on March 10, 2015, 09:31:46 pm
Also, how would I prove whether there is or isn't a sequence of five natural numbers s.t none are square free?
You can do it for k instead of 5 as well: Use the Chinese Remainder Theorem to find a sequence a, a+1, ..., a+k-1 such that (p_i)² divides a+i (i.e. a = -i mod (p_i)²), where {p_i | i} is any list of distinct primes.

18
General Discussion / Re: Mathematics Help Thread
« on: March 10, 2015, 06:06:11 am »
Quote from: Helgoland on March 02, 2015, 08:45:51 pm
Can anyone imagine what a non-finite CW complex dominated by a finite one would look like? I've been wrecking my brain for days, and I have no idea...

The best I can offer is http://en.wikipedia.org/wiki/Wall%27s_finiteness_obstruction . An explicit (but probably to complicated) example is given at the very end of http://www.math.harvard.edu/~lurie/281notes/Lecture2-Wall.pdf .

19
General Discussion / Re: Mathematics Help Thread
« on: February 06, 2015, 08:17:52 pm »
Quote from: Bauglir on February 06, 2015, 06:21:56 pm
Having an explicit inversion is an important goal for the thing I'm working on, which is a number generation scheme

Considering that exp, ln, arctan and such are also only computed numerically, why is not using Newton's method or some Taylor series also a legitimate way of inverting it¿ Sure, it won't be a function you can put in your average calculator, but that's really the only difference.

In other words: there is nothing special about exp or ln, they are not that more easy to compute than many other functions. They just happen to have names (for good but different reasons) and the others don't.

20
General Discussion / Re: Mathematics Help Thread
« on: October 25, 2014, 06:15:23 am »
So of topological spaces (there are others, e.g. of chain complexes, giving rise to what is called homological algebra). Then I would recommend Hatcher's Algebraic Topology (should be chapter 2). It has lots of exercises and is one of the more elaborate books I am aware of.
In case you want to look at chain complexes and such (the methods are over all of mathematics, especially algebraic topology and geometry), I know that Weibel's Introduction to Homological Algebra is fine, but be warned that this subject, while not difficult, can get quite lengthy. Due to its nature, reading lots of homological algebra at once may be a bit dull, a lot of people learn it simply "along the way". I don't have the book at hand to point out the most relevant chapters, but everything about homology of complexes is relevant, and taking a look at the chapter concerned with simplicial methods might be a good idea after having seen simplicial homology in topology.

As mentioned there are others, e.g. group (co)homology or étale cohomology, but I would advise understanding one of the more basic ones above first.

21
General Discussion / Re: Mathematics Help Thread
« on: October 24, 2014, 11:25:57 am »
Quote from: Helgoland on October 24, 2014, 07:52:17 am
Does anyone have pointers to a good introduction to homology? My prof went through the technical details at a rapid pace, and I failed to keep up - and now I'm lost. Badly.
What kind¿

22
General Discussion / Re: Mathematics Help Thread
« on: October 23, 2014, 05:32:46 am »
You could use any algorithm that works for reals as long as that algorithm didn't somewhere use that 2 \neq 0 or that you have a notion of being positive, with the added bonus that you won't encounter numerics. Especially, I don't see why LUP should not work. You should be able to solve your equation in O(n^2.376), n the dimension, or whatever better algorithm is already known.

23
General Discussion / Re: Mathematics Help Thread
« on: October 21, 2014, 02:07:40 pm »
You are working in IZ/2 aka IF_2 aka the field with two elements (can't better format that here, I used I to denote a second stroke both times): 2 is the same as 0 there, and thus 1 is the same as -1, too.

24
General Discussion / Re: Mathematics Help Thread
« on: October 20, 2014, 11:59:09 am »
Calculate the Eigenspace with any method you want; you essentially need to solve the equation a^i v = v, which is homogeneous and linear.

If you want to use that formula to calculate the number of p-loops, then you would factorize p first, then calculate a^d for all divisors d of p (use fast exponentiation; you can also do it iteratively), then find the eigenspace dimensions (Gauß' algorithm or whatever else that works), then apply the formula.

25
General Discussion / Re: Mathematics Help Thread
« on: October 19, 2014, 07:34:03 am »
If we assume that 2^{dim F(a^i)] is simply the number of fixed points of a^i, then the result as written follows simply by Möbius inversion.
Thus if we take F(a^i) to be the eigenspace of a^i corresponding to the eigenvalue 1, then everything fits.

Also, the proof and statement of that theorem is unnexessarily weird and long.

26
General Discussion / Re: Over 5 million Gmail accounts compromised
« on: September 19, 2014, 09:24:15 am »
Quote from: Arx on September 17, 2014, 02:19:39 pm
But wouldn't "@battery 20staple14 correct#horse" dodge a dictionary cracker? I have no idea how this works, so I'm genuinely curious.
Partially. But a good password cracker mixes things up. Also, while some things look more obscure that others, this is just a human thing. To measure the actual security you would use what is called "entropy": it essentially measures how likely that string is to occur by chance. But note that "by chance" does not mean "just roll a dice for each letter", but "how often does it appear in nature" (where "nature" is some setting you use, e.g. "all data on the internet"). Thus your dice show letters, but also words or other common sequences, each with their own probability coming from reality.

The string "battery correct horse staple" has pretty low entropy as it, as demonstrated by this thread, occurs quite often. The string "quantum dwarf house thorn" would be much better (but lost a lot due to now being mentioned here). For similiar reasons, using 100 consecutive letters from your book of choice is not much better than using 1000 consecutive letters instead: both appeared at least once, and probably the same number of times each, making them almost equally good.

On a similiar matter, adding just some numbers at the beggining and end of a password does not have any more effect than just adding one or two english words at the end, unless the added words are in a rather common position (e.g.: extending "crocodile ocean yellow" by "submarine" is probably worse than using "air", despite the difference in lengths).

27
General Discussion / Re: Wikipedia refuses to delete photo as 'monkey owns it'
« on: August 11, 2014, 03:29:31 pm »
My dictionary translated both the same. I already got your distinction the first time (I am not even doubting it) and I already told you to do the necessary replacements. No reason to be that insistent...

28
General Discussion / Re: Wikipedia refuses to delete photo as 'monkey owns it'
« on: August 11, 2014, 02:30:07 pm »
1. You gave this pedantic difference between value and price (which for me, a non-native speaker, doesn't even exist). If it makes you happy you can replace value by price everwhere.

2. This is just a strawman argument. My sole goal was to argue that information has a nonnegative value (or prize, whatever) which may decrease by sharing. I did that.
Now you claim that it also has to satisfy several other conditions. No, it doesn't. It is doing it's job just fine. You only argued that my example does not work if one wants to speak about long term effects (and some people lacking any imagination disputed it on the guardian being morally wrong, obviously lacking the minimal imagination necessary to replace "doom" by "loss of $100" or whatever) and such; yes, it does not work for that; because it isn't required to!
And thus those things are just nonsensical if applied to the example. Morals and such may be relevant to the monkey photograph, or to the question whether copyright is ok as is, but it is not for this one.

29
General Discussion / Re: Wikipedia refuses to delete photo as 'monkey owns it'
« on: August 11, 2014, 11:13:49 am »
Quote from: Loud Whispers on August 11, 2014, 07:52:43 am
In Britain a couple hundred years ago toll gates were made to charge people to use the roads to transfer goods to and fro the ports for trading. The people who owned these toll gates could get a nifty bit of profit for essentially being doormen to the roads. By the time the industrial revolution kicks in railways and canals render the turnpikes obsolete because the toll gates are inferior to both canals and railways and the inefficient regulation of the roads is proving to be hurting free trade.
In any case all the turnpike trusts went out of business, the middle man was eliminated and the locals celebrated by throwing the gates open, symbolic of their new found freedom.

The internet has done what the canals and railways of old have done and thrown open the intellectual gates of humanity to unending freedom and inane distractions; why do people like the random guy in your example earn themselves immunity to having to adapt to the changing world where no others have such a privilege? How is that a loss that demands the closing of the gates? Hell, I've got a friend who used to manufacture CD-ROMs. Key word here is used to, because he realized that no one was buying CD-ROMs anymore and if he didn't find another career soon he'd be made redundant; even more relevant is that CD-ROM sales are down because everything is digitized now. Should he have attacked people making digital purchases whilst clinging to CD-ROMs, or was he right to adapt?
Again: The whole example is one for the worth of information and its loss by sharing. It is not about any of the other stuff you want to insert into it; especially not about long term effects, morals or whatever nonsense people not reading it properly mentioned so far. I said that several times now and that's why I said that you should read those posts.

Quote from: Jelle on August 11, 2014, 09:27:19 am
Quote from: ZetaX on August 10, 2014, 08:49:44 pm
Machines can do a lot on their own nowadays and I do not see how an accident involving a falling meteor is any different than an accident of a cow running into your car. Both are out of human control, and if you want to argue like that you will have to explain why the monkey incident is any different.
Quite frankly it baffles me how you continue to compare living, breathing, thinking creatures to inanimate objects. An animals thinks, even if rudimentary in most cases, and acts; a man, who is also an animal, thinks and acts.
What is this, some kind of pre-version of that announced PETA flame¿ :P
Are you seriously raging about me comparing cows running in front of your car with meteors falling, as both are random events that could happen to anyone¿ How the heck should there be a difference between animal, machine or completely inanimate object if it comes down to something like a random event¿


Quote from: Jelle on August 11, 2014, 09:27:19 am
Claiming that any action involing anything not human can be reduced to some random natural event is honestly insulting.
Now you are claiming that saying that some events are random is insulting¿ What.

30
General Discussion / Re: Wikipedia refuses to delete photo as 'monkey owns it'
« on: August 10, 2014, 08:49:44 pm »
Quote from: Aklyon on August 10, 2014, 06:51:14 pm
Quote from: ZetaX on August 10, 2014, 06:25:21 pm
That's now just a non-sequitur. Shall I now give you an equally irrelevant answer about how you seemingly want total communism for communisms sake¿
No, but I can see you have a career in american politics with that kind of remark (minus the inverted ?). Copyright is an important concept as it originally was, modern copyright is a joke on par with software patents.
Perhaps future copyright will actually make sense for digital media, or more likely it won't, instead the entertainment industry and the IP lobby will continue to demand any new media form sacrifice their business and profits in the judicial system (or to unthinking algorithms that cannot distinguish allowed use from fair use nor illegal use from either of the above) so the industry never has look up and adapt further than it already grudgingly has.
I really don't get what weird problems you have. Neither did I agree to modern copyright in general, nor was I even talking about it in that post. How you got from a special instance we are discussing to such a general nonsense is beyond me. It seems you are just projecting some hatred of copyright onto me.


Quote from: Aklyon on August 10, 2014, 06:51:14 pm
Machines and accidents are different from other animals though. Machines do nothing on their own as a tool and accidents aren't a being of any kind, they have to have someone (or someone's something) to have an effect on.
Machines can do a lot on their own nowadays and I do not see how an accident involving a falling meteor is any different than an accident of a cow running into your car. Both are out of human control, and if you want to argue like that you will have to explain why the monkey incident is any different.

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