8,526 reputation
21038
bio website supermath.info
location Virgina USA
age 36
visits member for 2 years, 8 months
seen 6 hours ago

I'm interested in mathematics which is used to frame physical theory. I guess that means I'm at least a little interested in just about anything.

Currently I'm trying to understand Cartan's Method of moving frames as it applies to various classification questions of low-dimensional geometry.

I also have interest in superanalysis and certain problems of hypercomplex analysis.

More generally, I'm just looking for interested students who want to exceed the status-quo of undergraduate mathematics. Ideally, our interests overlap.


Aug
27
comment How hard is it to endow a $\textit{Spin}^{c}$ structure on four-dimensional manifolds?
I was browsing Friedrich's book earlier today. I think Branimir's comment is worthwhile. If you haven't looked at that yet it's probably worth a few minutes.
Aug
23
comment Solving a wave equation: $a^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$
The case depends on the boundary conditions. Different boundary conditions give you different $\lambda$ contributing nontrivial solutions. So...
Aug
23
comment Solving a wave equation: $a^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$
seems like you're missing the boundary conditions and perhaps an initial condition or two...
Aug
19
reviewed Approve Solution in terms of Lambert $W$ function
Aug
19
comment Exponential of a polynomial of the differential operator
@AnthonyCarapetis Thanks, guess my conjecture is dead. We'll just have to calculate it as Semiclassical indicated.
Aug
19
reviewed Reject first order linear PDE solving
Aug
19
reviewed Reject Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$?
Aug
19
comment Exponential of a polynomial of the differential operator
Let me attempt a guess here. The differential operator $aD$ acts on $x$ by $$ x \mapsto x+aDx = x+a.$$ Infinitesimally, $aD$ generates a translation. What does $g(D)$ generate infinitesimally? $$ x \mapsto x+g(D)x $$ To be explicit, let $g(D)=g_o+g_1D+ \cdots g_{n-1}D^{n-1}+ D^n$. Note $D(Dx)=D(1)=0$ hence $g(D)x$ truncates to $g(D)x = g_ox+g_1$. Thus, $$ x \mapsto x+g_ox+g_1 =(1+g_o)x+g_1. $$ So, I would conjecture (wildly) that $exp(g(D))f(x) = f((1+g_o)x+g_1)$ which of course reduces to Taylor's theorem in the case $g_o=0$.
Aug
19
revised History of the matrix representation of complex numbers
added 49 characters in body
Aug
19
comment History of the matrix representation of complex numbers
@JHance I thought this might amuse you. You're absolutely right. These $2 \times 2$ matrices just spring up in simple questions.
Aug
19
answered History of the matrix representation of complex numbers
Aug
18
comment Another way to solve this problem with complex expressions
@Semiclassical indeed, of course at the level of complex notation this is the $z \leftrightarrow w$ symmetry...
Aug
18
answered Another way to solve this problem with complex expressions
Aug
17
awarded  Tumbleweed
Aug
17
comment Some wedge product computation
@Berci his $w$ is a two-form.
Aug
17
comment Implementing trig functions for dual numbers
It's rather beautiful what you've shown here. Intuitively, $\varepsilon$ is "small". In view of that intuition, note that $\cos(v \varepsilon) = 1$ and $\sin (v \varepsilon)=v \varepsilon$ therefore, the result you derive follows from the standard adding angles formula for sine.
Aug
15
reviewed Reject Is there a name for the function $\max(x, 0)$?
Aug
15
accepted History of the matrix representation of complex numbers
Aug
15
comment History of the matrix representation of complex numbers
Fantastic answer. These notes by Wedderburn are nicely formatted, they should help me with some other hypercomplex history questions. That said, the paper by Cayley made me laugh. I assign a half-dozen of his points as homework problems in linear algebra. That paper should be required reading for my linear algebra class. "Convertible matrices= commutative matrices". It's interesting to see the analysis so tied to linear systems as opposed to the algebra of matrices themselves. In any event, I have to agree the quaternion passage implicits complex number rep. matrices. Thanks!
Aug
14
reviewed Reject Nested absolute operations