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Oct
11
comment Adjoint of Linear Maps
$T(V)=W$ is synonymous with surjectivity of $T$
Oct
11
comment Adjoint of Linear Maps
a linear map is injective iff it has zero kernel.
Oct
11
comment Is this Cayley Diagram contradictory?
Also, see weddslist.com/groups/cayley-plat/index.html for more cool examples of these.
Oct
11
comment Is this Cayley Diagram contradictory?
fwiw, I just noticed the Wikipedia article en.wikipedia.org/wiki/Cayley_graph contains the graph for $D_4$.
Oct
11
comment Prove $∇(∇^i a_n) = ∇^{i+1} a_n $
maybe some additional background to your question would be helpful.
Oct
5
comment Partial implicit differentiation question help - Solved
precisely the problem. Both $z$ and $x$ are functions of $x$.
Oct
5
comment Partial implicit differentiation question help - Solved
you need to think of $z = z(x,y)$ for this problem since apparently the problem assumes you should take $x,y$ as the independent variables.
Oct
1
comment On angles between subspaces in infinite dimensional spaces
It seems probable that the afterlife and the spiritual involves infinite dimensions. I wonder, is "angel" used in a technical sense somewhere in math? I know ghosts are no joke in physics. They are necessary for probability to make sense in certain quantized field theories if I remember the story right...
Sep
30
comment On angles between subspaces in infinite dimensional spaces
most excellent misspelling of title.
Sep
29
comment Finding the area of a parallelogram on a 3D coordinate plane
the cross-product gives a vector whose magnitude is the area of a parallelogram which takes the vectors as sides. Seems easy enough to find the vectors on the sides given the data you have...
Sep
22
comment Complex Analysis: Holomorphic Functions
nice. Still, I do wish more knowledge of the theory of real differentiability of functions from $\mathbb{R}^2$ to $\mathbb{R}^2$ was commonly known. It would make all of this far less mysterious...
Sep
21
comment Complex Analysis: Holomorphic Functions
btw, this answer is formal nonsense. Really, I'm just giving you a motivation for why the symbols $\frac{\partial f}{\partial \bar{z}}$ and $\frac{\partial f}{\partial z}$ are consistent with the usual derivatives of $x,y$. On the other hand, it does give us nice easy results like: $f$ is holomorphic if it is a function of just $z$ and not $\bar{z}$. A statement which on the face is really confusion since aren't $z$ and $\bar{z}$ dependent??? Well, no, if we merely use them as a complex notation for a pair of real variables.
Sep
21
comment Complex Analysis: Holomorphic Functions
I like to think of it in terms of the complex-linearity of the differential. If the differential allows us to pull out complex multiplication then that amounts to there being no term like $d\bar{z}$.... but, unless you've studied Jacobians and differentials this comment is probably useless (for now)
Sep
21
answered Complex Analysis: Holomorphic Functions
Sep
21
comment Complex Analysis: Holomorphic Functions
how do you define the partial derivative with respect to $\bar{z}$? When that is made clear, you can see that the given condition is merely a notation for the Cauchy Riemann equations.
Sep
21
reviewed Approve Let $\{E_k\}^{\infty}_{k=1}$ be a countable disjoint collection of measurable sets. Prove that for any set A…
Sep
21
reviewed Reject Is a certain kind of continuity needed on joint pdfs to switch order of integration?
Sep
21
reviewed Approve Probability Theory - CDF/Quantile Function question
Sep
21
reviewed Approve union of countable many positive sets has a positive signed measure
Sep
21
reviewed Approve Given a pairwise disjoint collection, $\limsup A_n = \emptyset$?!