14,373 reputation
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bio website math.poly.edu/~yang
location New York, New York
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visits member for 5 years, 2 months
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4h
comment Survey paper on isoperimetry
ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4
9h
comment Why do we teach calculus students the derivative as a limit?
Ben, the ones who got it wrong just guessed that the derivative formulas would be the same for degrees as radians. The fact that they guessed instead of making some effort to work it out was in fact what flabbergasted me. I would also note that many (maybe most?) mathematicians are uncertain about whether the constant is $\pi/180$ or $180/\pi$. This is not such a serious issue, but it bothers me that any engineer or physicist would be able to answer instantly.
2d
comment Beginners Guide to Cartan for Beginners
There are other approaches to this theory. A modern cohomological approach was developed by Spencer, Quillen, Guillemin, Sternberg, and Goldschmidt. A nice description of an involutive first order differential operator was given by Guillemin and is now known as Guillemin normal form. Using this description, it is easy to construct examples of involutive systems of PDE's. Still another approach, using only basic linear algebra, is described in the first chapter of my thesis, "Involutive hyperbolic differential systems", Memoirs of the AMS.
2d
comment Beginners Guide to Cartan for Beginners
Your question is still too vague. A tableau is essentially a way to represent a constant coefficient first order system of PDE's. So any arbitrary system of first order PDE's gives you an example of a tableau. You can then test whether the tableau is in involution or not and prolong if necessary. You need to say more about what you're hoping to learn about in the examples.
Dec
20
comment Beginners Guide to Cartan for Beginners
What's the question here?
Dec
16
comment System of linear first order PDE with constant coefficients
Finally, if the system is degenerate, then you should post it here, and someone can help you figure out what, if any, compatibility conditions are needed.
Dec
16
comment System of linear first order PDE with constant coefficients
You should check the references in the MO answer I linked to. They might handle the linear first order system case explicitly. Or you can compose your operator on the left by the cofactor matrix of your operator. This results in a higher order diagonal operator, which can then be analyzed using the results on higher order scalar operators.
Dec
16
comment System of linear first order PDE with constant coefficients
A compatibility condition can arise only if the system is degenerate. The system is non degenerate if there exists a nonzero $(t_1, \dots, t_n)$ such that the matrix $t_1M_1 + \cdots + t_nM_n$ is invertible. If this holds, there are no compatibility conditions, and the system has a solution for suitable $b$.
Dec
16
comment System of linear first order PDE with constant coefficients
You might try the answers to mathoverflow.net/questions/186779/…
Dec
16
comment System of linear first order PDE with constant coefficients
Could you say on what domain you want a solution? A bounded domain, all of $\mathbb{R}^n$, in a neighborhood of a point, or something else?
Dec
16
comment Moser's iteration for non homogeneous quasilinear elliptic PDE
And if you've spent many, many hours (or days) trying without success, you should post a more specific question explaining what you've been able to do so far and pointing out where you got stuck.
Dec
16
comment Moser's iteration for non homogeneous quasilinear elliptic PDE
Usually, any technique for analyzing PDE's has to be adapted to a specific situation. You should learn how Moser iteration works in principle and see if you can use interpolation estimates (such as the Gagliardo-Nirenberg inequalities) or whatever else you can find to make it work in this situation.
Dec
14
comment The definition of $W_0^{1,\infty}$
And Lipschitz functions are continuous.
Dec
14
comment The definition of $W_0^{1,\infty}$
For one thing $W^{1,\infty}_0 \subset W^{1,p}_0$
Dec
14
comment The definition of $W_0^{1,\infty}$
The answer is yes. And you still have trace equal to zero.
Dec
14
comment The definition of $W_0^{1,\infty}$
You get Lipschitz functions.
Dec
14
reviewed Close Interpolating (tangent)vectors on a sphere
Dec
14
reviewed Close Time estimate to determine if a number is prime
Dec
14
reviewed Close The singular value of $F(\theta)=\sin\theta\int_{-a}^{a}e^{-ikz\cos\theta}f(z)dz.$
Dec
14
reviewed Close On a paper by Yoneda