7,617 reputation
2832
bio website supermath.info
location Virgina USA
age 36
visits member for 2 years
seen 5 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.


1d
reviewed Approve suggested edit on Solution in terms of Lambert $W$ function
1d
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.
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reviewed Reject suggested edit on first order linear PDE solving
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reviewed Approve suggested edit on How many unique Binary Search Trees can be created with N keys?
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reviewed Approve suggested edit on How many unique Binary Search Trees can be created with N keys?
1d
reviewed Reject suggested edit on Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$?
1d
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 $g_o=0$.
1d
revised History of the matrix representation of complex numbers
added 49 characters in body
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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.
1d
answered History of the matrix representation of complex numbers
2d
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...
2d
answered Another way to solve this problem with complex expressions
2d
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 suggested edit on 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 suggested edit on Nested absolute operations
Aug
14
reviewed Reject suggested edit on If Sue buys A, she will have $\$1.50$ left; if she buys B, she will have $\$2$ left. Given that 2A=3B, how much money does she have?