If the fine structure 'constant' is a function of energy scale, then how is conservation of charge possible?

curiousdude · 13 days

That's why we have a classification system for differential equations. Some kinds of them have straightforward methods for finding solutions. Others don't.

RedDragon501 · 14 days

It's a difference of squares.

a² - b² = (a-b)(a+b)

vipercjn · 15 days

What a detailed question to look at.

Have you tried doing it in a different base?

avrus · 15 days

Just wanted to add that alpha is a function of many things in QED, and has actually been measured to vary. We tend to think its the charge that varies with it, but it really is indeterminate.

diffyqs · 16 days

Wolfram alpha does do some Laplace transforms, I think. Honestly, it is very easy. Get a book with a table and start substituting things in.

diffyqs · 16 days

There is, but it's actually extremely complicated compared to the Laplace transform method. And what I mean by t=0 is that the initial conditions all have 0 as their argument.

The other way of doing this... well you basically start by solving the homogenous part by finding a diagonal matrix that is similar to the matrix given by the coefficients of your system. You do this by finding eigenvalues and eigenvectors of your matrix. The eigenvalues become the exponents in your e^power*t , whereas the eigenvectors concatenate together to become a matrix that you then multiply your diagonal matrix by to get the fundamental set of solutions to your homogenous equations. The non-homogenous part is even crazier.

Maybe there is some other way, but I don't know it off the top of my head.

diffyqs · 16 days

Apply a Laplace transform to the system.

Do some algebraic substitution to get L(x) and L(y) as two different equations.

Apply the inverse Laplace transform to get the solutions.

(The trick to knowing that you can do a Laplace transform is if they give you initial values at t=0.)

peecatchwho · 16 days

Represent the sequence as a discrete function f(n). Say you have a converging function g(x) that intersects f(n) infinitely many times as n goes to infinity. Set g(x)=f(n) and find all points which match up, and you have a converging subsequence.

deepsoul13 · 17 days

In science, there is only approximate truth that relatively accurately describes a given set of situations. This might piss you off, but saying anything very general about reality is objectively and absolutely true is probably a bad idea, and impossible to prove empirically.

But, practically speaking, who cares? If a theory works as far as we can see, then it works.

deepsoul13 · 17 days

I would like to say that "I know that electrons are both particles and waves at the same time. I understand they are both always, never one or the other." is a false statement.

Consider the double-slit experiment. The properties of a quanta at any given time can be forced to become clear through measurement, forcing it to act either as a wave or as a particle, but not both. Whether they are both at the same time when not being measured is a very believable idea, though.

IAmVeryStupid · 19 days

Technically speaking, all you need to make it 'rigorous' is a set of consistent axioms that define it. Not that hard to do for full derivatives, actually.

As for the more particular questions, manipulating partial derivatives in the same way does not always work.

Take, for instance, this example:

Suppose z is given implicitely as a function z = f(x,y) by an equation of the form F(x,y,x)=0 This means that F(x,y,f(x,y))=0 for all (x,y) in the domain of f. If F and f are differentiable, then we can use the Chain Rule to differentiate the equation F(x,y,z)=0 as follows:

D(F,x)D(x,x)+D(F,y)D(y,x)+D(F,z)D(z,x)=0

Where D(a,b) is the partial derivative of a with respect to b.

So D(x,x)=1 and D(y,x)=0

so the equation is

D(F,x)+D(F,z)D(z,x)=0

If D(F,z) is not 0, then

D(z,x)=-D(F,x)/D(F,z)

If you were to treat this like a fraction of the form

dz/dx=-(dF/dx)/(dF/dz)

Then dz/dx=-dz/dx... Which restricts dz/dx=0, which is generally not the case.

I once made a rigorous system of differentials for real-valued functions. I had to modify the equivalence relation to make it such that it couldn't say dx=0, though. Other than that, it worked smoothly. The restrictions on the real-valued functions were only that the function be differentiable to the order that you're doing in differentials.

jcmc2112 · 20 days

They aren't 'just' a vector space. They are a vector space and much more.

strategyguru · 24 days

http://en.wikipedia.org/wiki/Principle_of_least_action

In this particular case, it is simple. You want the angle between the line and AC to equal the angle between the line and CB.

cliambrown · 25 days

Of course I believe that many things are possible. I don't see how you could say that something isn't possible just because we haven't seen it yet. I imagine that you would have been flabbergasted by electricty during the 18th century.

Symbols are tied to what they represent. That's the point of symbols. If you don't have symbols, you cannot express anything. Go ahead, try to tell me something without telling me something.

cliambrown · 25 days

http://en.wikipedia.org/wiki/Category:Physics_theorems

Um. I don't know what to say? You don't seem to know what you're talking about.

Specifically for 8, I don't know how you could interpret such a thing. If something is possible in the math, then it's possible within the theory. If the math accurately describes nature, then it is possible in nature. It doesn't have to be mandatory, but I wouldn't underestimate the universe's intricacies.

Also, theories /are/ constructed with language. If you don't have a group of symbols saying something, then you can't have a theory.

cliambrown · 25 days

Within classical EM, we have Maxwell's equations. Within Fluid Mechanics, we generally defer to Navier-Stokes. Within GR we have EFE. These are the 'axioms' you add on. You also take on a mathematical body to specify them.
That's only in some contexts. In proof contexts a hypothesis is a condition imposed upon a system. I'm sure you're familiar with words having more than one definition?
Lol. That's blatantly false. Consider Bell's theorem. Physics has a theoretical branch for a reason: physics is an outgrowth of thinking logically.
Physics isn't just about explaining things, but about predicting things. When you construct an equation that you are confident in (ostensibly because you've performed some tests on it), then you can use it to predict things.
Tell me, in precise terms, what is GR without EFE? Give measurably precise predictions based on curvature of space without them. Go on.
That's just your particular semantic definition of theory. Theory has other definitions.
The problem could just as well be that we don't have measuring tools capable of seeing them, or we don't know where to look. Nobody has data about what goes on in a black hole.
I don't think you understand the intricate relationship between theoretical physics and experimental physics. Have you heard the story on how antimatter was predicted and then discovered?

cliambrown · 25 days

In theoretical physics we have theories, which are constructed of theorems. We make our theories as close to observation as possible, and then spread the theory. A hypothesis is a condition used to prove a theorem. So we take Einstein's field equations and then construct a theory using them as hypotheses (or axioms, if you will).

The set of 10 PDE's that make up the constraints of General Relativity upon gravity and space-time allow for many different theories. Some of them allow for wormholes.

A wormhole would give you a mechanism for time travel. Who doesn't want to know more about that?

cliambrown · 25 days

And theoretical. The problem is how to measure it.

rymmen · 25 days

Yeah, but then you can remove the Riemann sphere while still being consistent when talking about asymptotic infinities, and further define operations on them. Very easy stuff, I agree, but still not obvious to most, as evidenced by people's responses here.

cliambrown · 25 days

So you define 'theoretical' as... what? Theoretical is the math to me. So wormholes are both theoretical and hypothetical (experimentally) to me.

The problem is there is no experimental data on what goes on in a singularity. So all discussion of such is theoretical.

rymmen · 25 days

Not at all, "greater than" is really only used in very specific contexts, when you're dealing with orders (or more generally partial-orders). From the context it should be apparent which order you're referring and the relation should only ever be applied to two elements of the sets the order is defined on. If none of these things are clear you should make sure to state them. This is how maths works. If you want me to however I will happily define greater than.

Define greater than. I will then ask you define any other mathematical term in your definition, and repeat this process indefinitely. Infinite regress is the problem of not having 'primitive notions'.

Things are clear to me. I mean, is 2+2=4 clear to you? When I say that the line y=x goes towards a specific infinity, this is also just as clear to me. When I say that a Riemann sphere has all the infinities of the complex plane coalesced into a single point on top of the sphere, that also is perfectly clear to me.

Now we have your case. First off "infinity" means a lot of things to a lot of people, it can be a point in the extended real number line, a point in the extended complex plane (or Riemann Sphere),it can be an ordinal, it can be a cardinal... etc

True true. Context dictates which is which, especially when the people you're talking with don't have any prior exposure.

Secondly the differential operator can be defined differently in many different contexts depedning on the objects it's acting on. Normally (as in the example of greater than) what you mean by the notation d/dx is apparent from the context of what it's acting on. However because there's obviously some confusion with your use of "infinity" you should state all this clearly. Even if you weren't to anticipate this (which you really should have), the correct response is to clarify your notation, not insist it is obvious.

Well, what is not obvious about infinite values being constants? If a and b are constants, then an operation of a and b is a constant. So z/0 (defined with the Riemann Sphere, where z is a constant complex number) is a constant. The derivative of a constant is 0. So d(infinity)/dz is 0.

You can move on to talk about separate infinities on a plane, independent of the Riemann Sphere, which is what I did.

As a final note, as far as I can make out this is not an interesting question/observation anyway, however you manage to frame it.

To you. The reason why I'm thinking about this is because Quantum Electrodynamics has asymptotic infinities that are problematic.

cliambrown · 25 days

It is saying... in the theory of general relativity the equations allow for the existence of a hypothetical wormhole.

So it is a theoretical possibility within GR.

cliambrown · 25 days

You said it didn't make it to theoretical. Which is false.

cliambrown · 25 days

There is no observational evidence for wormholes, but on a theoretical level there are valid solutions to the equations of the theory of general relativity which contain wormholes.

Read your own article.

rymmen

TROPHY CASE

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