/sci/ - Science & Math

I find this to be a fairly sweet feature and
>>5415941
explains how it happens. I'd add that it has a clear visualization too:

http://www.wolframalpha.com/input/?i=Plot[{Log[x]%2CLog[3x]}%2C{x%2C-1%2C2}]&dataset=

http://www.wolframalpha.com/input/?i=Plot[Table[Log[n*x]%2C{n%2C-.3%2C4%2C1}]%2C{x%2C-1%2C2}]

Namely the logarithmic function is just the function, with the following property:
The family of functions you get out of it by squeezing the x-axis equals the family of functions you get by shifting along the y-axis.
That means if you re-draw the function after scaling the x-axis via
<span class="math">x \mapsto a*x,[/spoiler]
<span class="math">log(x) \mapsto log(a*x) = log(x) + c_a[/spoiler]
(where a is some number)
the function you then get is just the previous log function shifted along the y-axis by some number <span class="math">c_a[/spoiler].

The point I'd like to make is that, in principle, there is nothing all to special about the log function here, except for the transformation
<span class="math">x \mapsto a*x,[/spoiler]
appearing natural to as because it's linear!
More elaborate, given a function <span class="math">f(x)[/spoiler], we can compute the transformation of the x-axis
<span class="math">x \mapsto T(x),[/spoiler]
which will result in a shift
<span class="math">f(x) \mapsto f(x)+c_T,[/spoiler]
Namely we have to solve the differential equation.

<span class="math">f(T(x))'=f'(x)[/spoiler]
or
<span class="math">T'(x)=\frac{f'(x)}{f'(T(x))}.[/spoiler]

If you plug in <span class="math">f(x)=log(x)[/spoiler], you'll find <span class="math">T'(x)=T(x)/x[/spoiler] so that T(x) is just the squeeze <span class="math">T: x \mapsto a*x[/spoiler].

cont.

Erdos number 4 and Bacon number actually 3.

Ten year ago is a bit to far back, but things I persoanlly discovered for me and which I find helpful:

- Get yourself interested in/learn about stuff _before_ you have to learn them in some class
- Read a lot
- Write down things you understood in a way your friends can understand them
- Write down = Formular what seems incomplete/strange/ununderstood to you (maybe if learn more about it reformulate why things don't add up, stick with the question until it's resolved.)
- Find out what the axioms are and (in physics) what it is what the theory is supposed to do (this is often not always so obvious)
- Don't be afraid to ask questions (=try not to not ask because you want to be thought of as clever)
- Having someone who knows more than you is always good
- I always like to have a couple of problems/projects going on, so if I feel I can't do something I switch to the other thing
- I try to connect with other subjects I like and try to connect the dot's
- Try to understand why people like certain things/theories/subjects... after you deal with them for a while, you end up liking them too
- ...


Clearly this is a general learning behaviour description, not a career advice
(I don't know where I am in 3 years, ...taking the developement of at least the last 10 years into account, given enough information from academics, there seems to be no good reason for me to go the classical academics path)

I have a masters in physics, working on a PhD, but I feel that I really want to study something else afterwards. Like ...linguistics.
It's a money question of course. Throwing away some time of education, so to speak.

>>4723991
>It would be amazing if this board become a hybrid between discussion-oriented physicsforums and question-oriented stackexchange with a laid back aspect to it. Honestly, it would be possible with more moderation. We still get career advice/popsci crap/troll threads on here hourly. I never see anyone discussing a recently published paper anymore.

You'd need people for that. If you just earase the shitty posts, there are still hardly enough users to fill up a board of interesting topics

>>4599827
>You're a smart ass retard that like talking out of your ass
Why do you insult me?
>Question of semantics and syntax are the same with Gödel's completeness theorem and soundness so fuck off with the formality when they don't matter.
OP was asking a question within Predicate Logic and was using a specific set. I was pointing out to the second poster that 'true' or 'false' are relevant in semantics, not in pure logic. The purpose was to find out why he seems to associate the notion of an empty set with truth or falsehood. I don't see how you've cleared that up. Truth or false are relevant concepts when it comes to Gödels theorems, as you rightly say. "empty set", "no set", that has a priori not to do with falseness.

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
It exists in set theory - true. In Zermelo–Fraenkel set theory it comes from the axiom of infinity
The empty set is not a priori to be found in predicate logic.
http://en.wikipedia.org/wiki/Axiom_of_infinity

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
Also we practically never EVER care about how the underlining ordered pair structure is made in proofs, (a,b) could just as easily be {{a},{a,{b}}}. >And we never care about the set theoretic construction of the natural numbers unless you want to write {0,1} lazy as 2.
I was just constructing models to show the use of the constructions in OPs post. This was the answer of the question. The answer is "yes, in a set theory context, these constructions are valid."

>>4524663
But you really only desire such a theory, because of interpretations of physics, don't you.
I think "make a lot more sense if we assume that space is full of some fluid" are very problematic, because you're just extrapolating the experiences of a world more close to you, more macroscopic, to the microscopic world.
I mean I'd also like the things to be in a way I can simply grasp and understand them, but what I want or what I find reasonable ("make a lot more sense") will not have any influence on how nature is coming out to be.
As far as physics is concerned, nothing has "to make sense", other than that you want your mearureable observables to be represented in the theory, and that it's mathematically self consisted.
"shut up and calculate".

It's the potential, which is "natural" in S and P.

It's useful if these (S and P) are the parameters, which are constant in the process you're interested in, because of course then the function doesn't change in these. If you know that P=const., then dP=0 and hence

dH=(∂H/∂S)dS=TdS=dQ.

I.e. the enthalpy change is the heat flow.
A typical situation where P=const is when you consider chemical reactions at free are/laboratory, becuase then the preasrue is just 1atm, whatever happens. An interesting such situation is aerospace physics, i.e. combustion in a turbine.

On a broader note, the enthalpy H(S,P) is a natural potential, while U(S,P):=U(S,V(S,P)) is not. The problem is that the variable P is derived from U via a derivative P:=-(∂U/∂V)_S, and if you wanna use such a construct as a variable you might lose information. You can use U(S,P), or U(T,V) or other variant, but if you're just given these on the spot, you can't really reconstruct the true system.
The Potentials, i.e. U(S,V),S(U,V),H(S,V),F(T,V),G(T,P),... really contain the full information because they aren't just substitutes from U, but are derived via a Legendre transformation, which is designed to make a substitution, where one variable is a derivative.
An examples from mechanics is the Hamiltonian L(v,q) --> H(p,q), where q is space, v velocity, and p the momentum, which really just is proprtional to ∂L/∂v.