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March 11

Existence and uniqueness of solutions

Consider the differential equation $y{\frac {dy}{dx}}+(1+y^{2})\sin x=0$ . Discuss the existence and uniqueness of solutions without solving the equation.
I have been given a theorem which states that, given ${\frac {dy}{dx}}=f(x,y)$ , if $f$ and the partial derivative of $f$ with respect to $y$ are defined and continuous throughout a rectangle R containing a point $(x_{0},y_{0})$ in its interior, then there exists a unique solution around $x_{0}$ . Is it okay to rearrange the given differential equation to ${\frac {dy}{dx}}=-{\frac {(1+y^{2})\sin x}{y}}$ and then use this theorem? i.e. there exists a unique solution at all points where $y\neq 0$ . Widener (talk) 12:20, 11 March 2012 (UTC)[reply]

There seems to be something missing in the statement of the theorem, perhaps the addition of an initial condition (maybe that the solution must contain the point (x₀,y₀)). As a counterexample, take for example where f(x,y)=1, the general solution to dy/dx=1 is y=x+c, c∈ℝ. This is a family of solutions, not a unique solution, but fits the prerequisites of the "theorem". — Quondum ^☏✎ 13:32, 11 March 2012 (UTC)[reply]

That sounds probable. Widener (talk) 13:37, 11 March 2012 (UTC)[reply]

It is hard not to solve the equation!

Separation of variables:

{\frac {ydy}{1+y^{2}}}+\sin xdx

is zero.

Doubling and integrating:

\log(1+y^{2})-2\cos x

is constant.

Taking the antilogarithm:

(1+y^{2})e^{-2\cos x}

is constant.

Bo Jacoby (talk) 19:33, 11 March 2012 (UTC).[reply]

Is your theorem the Picard-Lindelöf theorem (this assumes Lipschitz continuity, but I think that is implied by continuous derivatives)? Anyway, I think it is fine to rearrange the ODE, as long as the new ODE satisfies the conditions of the theorem. I suppose you would need to deal with the

y_{0}=0

case separately, though. 130.88.99.218 (talk) 18:01, 12 March 2012 (UTC)[reply]

I think you would need to solve the differential equation in order to investigate the case

y=0

Widener (talk) 21:43, 12 March 2012 (UTC)[reply]

Quantum mechanics for mathematicians

Is there any book or source that treats quantum mechanics from a mathematician's perspective? I've read several physics textbooks on the formalism in terms of linear algebra and when they got to the Dirac delta I had no idea what's going. Money is tight (talk) 13:47, 11 March 2012 (UTC)[reply]

You may be interested in the Wightman axioms, constructive quantum field theory, or Hilbert's sixth problem. Widener (talk) 14:15, 11 March 2012 (UTC)[reply]

I'm not sure at what level you're aiming for, but this book has an account for mathematicians who are absolute beginners in the subject: http://www.math.cornell.edu/~gross/Brian-Hall's-QM-book.pdf There are many other books targeted at a more advanced level such as Prugovecki (who gives a rigorous introduction to the Hilbert space background, but I don't think you can get much quantum mechanics out of it unless you know some going in). Sławomir Biały (talk) 14:42, 11 March 2012 (UTC)[reply]

Thanks!!! Exactly what I needed. But it seems to be missing some sections, especially the sections in the last chapter. Do you have a complete version? Money is tight (talk) 02:20, 12 March 2012 (UTC)[reply]

Try this: [1] Sławomir Biały (talk) 14:49, 12 March 2012 (UTC)[reply]

Dimensions in vector spaces

A lecturer I'm studying with claims that if ${\mathcal {O}}_{L}$ is an integer ring (an integral domain) and ${\mathcal {P}}$ is some prime ideal in it, then $\dim({\mathcal {O}}_{L}/{\mathcal {P}}^{e})=\dim({\mathcal {O}}_{L}/{\mathcal {P}})+\dim({\mathcal {P}}/{\mathcal {P}}^{2})+\ldots +\dim({\mathcal {P}}^{e-1}/{\mathcal {P}}^{e})$ . Why is this? I'm probably being stupid but shouldn't the relationship be something more like the tower law? Shouldn't we get an overall basis something akin to the product of elements of intermediate bases, rather than having a sum of dimensions as is claimed here? I think I'm being stupid but I can't quite see why. 86.26.13.2 (talk) 14:20, 11 March 2012 (UTC)[reply]

If you have a module V with a submodule W, then dim V = dim W + dim (V/W). Here

{\mathcal {P}}^{e-1}/{\mathcal {P}}^{e}

is a submodule of

{\mathcal {O}}_{L}/{\mathcal {P}}^{e}

, and the quotient is

{\mathcal {O}}_{L}/{\mathcal {P}}^{e-1}

so

\dim({\mathcal {O}}_{L}/{\mathcal {P}}^{e})=\dim({\mathcal {P}}^{e-1}/{\mathcal {P}}^{e})+\dim({\mathcal {O}}_{L}/{\mathcal {P}}^{e-1})

. Then repeat. Rckrone (talk) 22:40, 11 March 2012 (UTC)[reply]

For a motivating example consider

{\mathcal {O}}_{L}=\mathbb {Z}

. Any number a in Z/p^e can be uniquely expressed as a = a₀ + a₁p + a₂p² + ... + a_e-1p^e-1, with 0 ≤ a_i < p. Rckrone (talk) 22:52, 11 March 2012 (UTC)[reply]