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Old 09-24-2014, 09:42 AM   #9881
JamSessionFreak
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Quote:
Originally Posted by sickman411
Ah, right. You made a small mistake when you applied the logarithms, and I hadn't noticed. Don't forget that only x is raised to the power x, not 2. The expression you have there is for y = (2x)^x, not y = 2 x^x

Here, I couldn't be arsed to type it all out and wouldn't look as neat anyway.

[IMG]solution.png[/IMG]

Yeah, this is spot on, no point in me posting the same thing.
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Old 09-30-2014, 12:33 AM   #9882
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Probability theory question:

Suppose that three numbers are selected one by one, at random, and without replacement, from the set of integers {1, 2, 3, . . . , n}. What is the probability that the third number falls between the first two if the first number is smaller than the second?
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Old 09-30-2014, 01:46 AM   #9883
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There are n(n-1)(n-2) different ways to select three numbers out of that set.

Suppose n_1 is the Nth number and n_2 is the Mth number, then there are only M-N-1 number choices left for n_3.
Sum over all M, N for M > N > 0, M < or equal to n.

Divide by n(n-1)(n-2).

edit: actually that completely ignores that the first number is smaller than the second one. So, what you want is P(*that event* | n_1 < n_2) and that is P(*that event* AND n_1 < n_2)*P(n_1 < n_2)

P(n_3 falls between n_1 and n_2) is what I found, now you need P(n_1 < n_2).
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Old 10-02-2014, 02:42 AM   #9884
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So I'm starting to read about Homotopy Type Theory.

HELL YEAH
**** YOU SET THEORY! NOW I CAN STOP HIDING MY HATRED FOR YOU MWAHAHAHAHA.

Oh, sorry I got a little carried away


Anyways, is there a "General Math" thread? Because as much fun as me rambling to myself is, it doesn't seem that productive.
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Old 10-02-2014, 05:04 AM   #9885
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I just wanted to share with you this website : https://www.khanacademy.org/
It's a mazing site to learn mathematic.
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Old 10-02-2014, 12:28 PM   #9886
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Gonzaw, coincedently, I recently started getting into type, category and proof theory. Did you already saw the Steve Awodey (HoTT author) lectures on category theory on YouTube? They're part of some foundational series on foundations of computer science, logic and type theory. They're amazing. I'm starting now with that series. Pierce's book "type theory and programming languages" is next. Then I will try to tackle HoTT.

SET THEORY IS BRAINWASHING

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Old 10-02-2014, 07:39 PM   #9887
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Originally Posted by niqolaise
Gonzaw, coincedently, I recently started getting into type, category and proof theory. Did you already saw the Steve Awodey (HoTT author) lectures on category theory on YouTube? They're part of some foundational series on foundations of computer science, logic and type theory. They're amazing. I'm starting now with that series. Pierce's book "type theory and programming languages" is next. Then I will try to tackle HoTT.


No, I haven't. Thanks, I'll check that series soon.


Here's an interesting article:

https://golem.ph.utexas.edu/categor...ype_theory.html

This particularly caught my eye:
Quote:
The solution to this problem is the univalence axiom, which can be concisely described as saying that “isomorphic structures are equal”. More specifically, it says that if A and B are types, then proving that A=B is the same as specifying an isomorphism between A and B. This implies that analogously, structured types, such as natural numbers objects, groups, rings, fields, etc., are equal if they are isomorphic by an isomorphism preserving the structure


I think this is what separates homotopy type theory, with any other normal type theory (Martin-Löf kind of stuff) from programming languages. You can have identical types, which are actually different, but isomorphic in their structure.

I haven't really thought about that before, but it makes so much sense. HOW you populate the elements of the type, shouldn't forcefully define WHAT that type is. I see this ALL the time with programming languages, where you define a "List" as either having pointers, or being a "Null+Cons X List" algebraic type, or being something else.
They are all different types, with different structures too, but we all treat them the same. In programming, if someone where to define "List" as an interface/abstraction, he doesn't care how you implement it (i.e what the actual type is), he only cares that it is a list. The isomorphism between those structures is what makes that type a List.

The beauty is how it applies to mathematics in this case. You can define the Natural Numbers, Real Numbers, etc, in ANY way you like. But as long as the essence of them is the same (i.e you can create an isomorphism between them), then it doesn't matter. This seems crazy! Yet....isn't this how we actually think and talk about the world?

We never talk about natural numbers as what they represent in Peano arithmetic, nor how they are constructed in ZFC ("So Sally, if I have 10 dollars, and 10 belongs to the set 16, then how many dollars left do I have?" nobody talks like this). Yet we all talk about the same thing. The same happens with real numbers. I still don't understand how real numbers are defined or constructed in ZFC, but I don't care, I already know what they are, and in my mind they have a certain structure that is isomorphic to the one defined by ZFC.


This brings me to an important question: What are the equivalence classes of this isomorphism?
If we define ZFC natural numbers as Nat1, and Peano numbers as Nat2, we can find an isomorphism between the two. If we use this as an equivalence relationship, then Nat1 ~= Nat2 (and in homotopy type theory, both are identical types).
So if this isomorphism defines an equivalence relationship, then it has equivalence classes. What are ALL these classes? I guess they entail all the possible "different" types out there, right?
Or.....do they define all the possible mathematical objects out there? Natural numbers would be one class, integers another one, rational numbers another one, etc. Is there a way to find all of these, so we can find ALL possible mathematical objects (even the ones we haven't come up with yet)?. Does this also mean I can write a random type in a programming language, and this will 100% be isomorphic to a real mathematical object, and thus I can work with it as I would that mathematical object?
Can we automate this "find the isomorphic mathematical object for this type" process? This way we can just create types that are convenient to us (when trying to solve a particular problem, or trying to prove a particular theorem), resting easy that the already ARE the object we want to really talk about.


Anyways, I wish there were actual programming languages with this in them. Not added as a different part of the language (like "interfaces", which is basically what that univalence axiom represents in programming languages), but as part of the creation of types itself. If I create type "NaturalLOLNumber", and I create this specific isomorphism with "Natural", then automatically allow me to use it with "Natural" interchangeably.
Not even "strongly typed" programming languages have this kind of stuff, to do this kind of stuff you have to create ugly functions with ugly type signatures. Case in point, Haskell with typeclasses. You want to create a function "addNatural :: Natural -> Natural", but instead you are forced to create a "addNatural :: (Natural n, Natural m) => n -> m" one. It's more....natural to specify it like the former (pun intended)


Anyways, this is fascinating. Finally I can start loving math again.

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SET THEORY IS BRAINWASHING


YES! BURN! BURN THE SET-RETICS!!
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Old 10-02-2014, 08:15 PM   #9888
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My second question would be: Do these mathematical theories actually matter?

We treat ZFC, ETCS, homotopy type theory, etc, as equivalent. They all "define" maths. Maybe we could come up with other ones too. It's like we can change them as we want, and the "foundation" of maths will be the same. Real numbers will keep being real numbers, and we can still prove the same theorems for them, depending on which one we use (in a general sense).

So, what exactly makes them so "disposable"? Aren't they what actually define what math is? So how can they define math, and be disposable at the same time? Is there something "else" that defines math, and we just use these theories to "bring it down to earth" so we can work with it? Something ala Plato's ideas or something.
This is similar to the question about the Natural numbers and stuff. WHAT makes the natural numbers what they are? They can be encoded in different theories, and in homotopy type theory even in different types, but they are all the same and refer to the same natural numbers. They never changed. Why is that so? And how do you arrive at such a conclusion about the thing we call "natural numbers", and not about any other random thing we can randomly point at?

EDIT: Is there a metric for defining what the "optimal" mathematical theory would be?

Something like "Oh, I can work with the set/group of natural numbers, and the powerset of them, and the powerset of that too, with very few problems! This is a +5 for me!", but for every "important" part of maths you may wonder about.
So you use this metric to compare the theories, and can say which one is the "better" one.

If this metric exists, the obvious question is: What is it? But you can find more questions. For example, is the "optimal" theory attainable? Or is it impossible to find, and we have to be content with the theories we have, and if we DO create new theories, they will be "worse", or at best "equal" than the ones we have now?
If there is no optimal, can we at least be able to consistently create better and better theories? Like, making the "optimal" theory an asymptote we try to get closer and closer to?
Is there any way to do any of this at all, or is it "undecidable" from the beginning, for some reason?

Last edited by gonzaw : 10-02-2014 at 08:20 PM.
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Old 10-04-2014, 12:25 AM   #9889
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Wait a minute....
http://en.wikipedia.org/wiki/Univalence_axiom

Does this mean that Natural numbers and Rational numbers are the same in HoTT?

wat
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Old 10-07-2014, 05:36 PM   #9890
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What happens when H+ reacts with CO3 2-?

Ofcourse h2co3 will form. But in what state is the h2co3? Is it whole molecules of H2CO3 dissolved? (Ignore the fact that h2co3 decomposes to h20 and co2) or will it split back into H+ and CO3- thus making the process a constant back and forth equation? (don't know what it'scalled in English).
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Old 10-07-2014, 06:19 PM   #9891
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Quote:
Originally Posted by liampje
What happens when H+ reacts with CO3 2-?

Ofcourse h2co3 will form. But in what state is the h2co3? Is it whole molecules of H2CO3 dissolved? (Ignore the fact that h2co3 decomposes to h20 and co2) or will it split back into H+ and CO3- thus making the process a constant back and forth equation? (don't know what it'scalled in English).


The "back and fourth" is "equilibrium."

H2CO3 <-> H2O + CO2 is what happens in solution. The concentration of anything else in solution is so tiny that it is extremely difficult to measure. For all intensive purposes, H+ + CO3 2- is really CO2 + H2O.
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Old 10-08-2014, 02:22 AM   #9892
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Originally Posted by DamienEx1021
The "back and fourth" is "equilibrium."

H2CO3 <-> H2O + CO2 is what happens in solution. The concentration of anything else in solution is so tiny that it is extremely difficult to measure. For all intensive purposes, H+ + CO3 2- is really CO2 + H2O.

So there are some points where there is an actual complete molecule of H2CO3?
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Old 10-08-2014, 09:03 AM   #9893
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Originally Posted by liampje
So there are some points where there is an actual complete molecule of H2CO3?


Here is the deal. Carbonic acid, by itself with no water, will exist extremely stable as H+ and CO3 2-. Once you introduce water into the system, it catalyzes the decomposition of H2CO3 into H2O and CO2. This will typically leave behind a single carbon forming a bicarbonate HCO3- with dissolved CO2 in solution.

At some point, yes there is a H2CO3 if there is no water, but since the equilibrium constant severely favors the decomposition, and most of the time you are dealing with carbonic acid in water, H2CO3 does not exist in solution.


I am curious, what class is this for? Every chemistry class I took was focused on the fact that it breaks down in solution and you never have to deal with H2CO3. This is by no means an easy topic. You could almost have a semester on the behavior of Carbonic Acid.

*Edit* I did a little more digging for a better explanation. If you have a single molecule of H2CO3 in a vacuum with no water present, it has a chemical half life of 0.18 million years. The moment you introduce a second H2CO3 molecule or add water into the system, it canalizes the reaction and breaks down into H2O and CO2.

In a typical solution where you have CO2 dissolved in H2O, it will form H2CO3 very slowly and immediately decompose into H2O and CO2. The ka value is as follows:

[H2CO3]/[CO2] ≈ 1.7×10^−3

This significantly favors CO2 in equilibrium.
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Old 10-08-2014, 03:42 PM   #9894
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It is not really for a class. I was busy with acid-base reactions a while ago. And I always wondered why H2CO3 is the product of H+ and CO3 2-. And why, even though it is in water and decomposes, you write it down as H2CO3.

I'm surrounded by language barriers rignt now. But It's purely for own interest.
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Old 10-08-2014, 03:54 PM   #9895
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Alright, that is actually helpful in explaining this then. The only reason it is written as H2CO3 is because it can exist in that state in perfect conditions. Under perfect conditions you could have H2CO3, but since that never happens you will always see it written as:

H+ + CO3 2- -> CO2 + H2O

People who don't know any better will write H2CO3 and they would be wrong. Common mistake.
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Old 10-08-2014, 04:06 PM   #9896
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I've very rarely seen it written as H2CO3.
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Old 10-08-2014, 04:38 PM   #9897
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I've very rarely seen it written as H2CO3.


I've only seen it as H2CO3 as an intermediate or when writing the ka for equilibrium constants.
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Old 10-08-2014, 06:21 PM   #9898
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We learn to write it down like this:

2HCl + NaCO3 -> Na + +2Cl- + H2CO3
Followed by
H2CO3 -> H2O + CO2

I guess I just learn how to do it the most systematical way followed by the exception.

Edit: It's also easier to make the reaction with the right amount of molecules. Thus making it easier to calculate moles and such.

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Old 10-08-2014, 06:36 PM   #9899
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Is anyone here good with calculus? Lagrange multipliers specifically, if so then I think I'm going to be asking for a bit of help at some point soon. I'm getting the basics but its really pushing my skills to the limit and I'm finding it very difficult
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