@Paddy $z_1^2+z_2^2$ is a multivariable function with a complex gradient. If you want a function on $\mathbb{R}^n$, then you will need some complex coefficients, like $ix_1+x_2$. I'm not sure what you are looking for.
Of course a real gradient is also a complex gradient ($\mathbb{R}\subset\mathbb{C}$).
When two functions intersect at a point does the slope of the two functions at the intersection point have to be different or can these also be the same?
I had an elective on writing systems of the world and as a part of that course, we were given some Japanese pens and asked to write hiragana/katakana and I found writing that very difficult.
in that course of ours,you have to take care of how you draw every line (from Right to left or left to right, from up to down or down to top and strokes etc.)
i'm left handed, so i make a lot of letters in an unconventional way. learning cursive was very difficult for me because the whole alphabet is premised on you being right handed.
it's all about clockwise movement. every letter and letter feature is oriented.
or counterclockwise movement. i mix them up because i'm left handed.
so the way to summarize it would be that, if you're going to assign any value to this series, it'll be $(\sqrt{5}-1)/2$. but said assignment will not be summation as such, because the series doesn't converge
to review: form all $\pm$-tuples of $(A,A',B,B')$, and form the 8-tuples $(A,A',B,B',AB,AB',A'B,A'B')$. Take the convex hull of these points to get 7-dimensional polytope
> @schn it all depends on the application to which it is being applied. There is no way to make a generalization that will be true in every case. Some cases require a uniform convergence and some don't.
Dear @robjohn , can you give an example of a case where uniform convergence is required and possibly another where it isn't?
Are there specific rules for when constants can and cannot be lumped while solving differential equations? When working with ODEs, I operated under the heuristic that you cannot lump once you start the process of applying initial conditions. Now when working with PDEs, it seems like sometimes you have to lump in the middle of applying initial/boundary conditions, but sometimes doing so seems to lead to a wrong answer.
The gradient of $||x||^2$ is $2x$ for a real vector $x$. This doesn't seem to be true for complex vectors. Any idea what the gradient will be for complex vectors?
So I messed around a little with the math and came to the conclusion that $x + x^*$ is the gradient in general and this would work for a complex vector too. I passed this to a gradient-check function and it confirmed that the gradient was right.
BUT
I am no mathematician. Engineering grad student here
If a function has a gradient $g$ at a point $x$ then $f(x+h) = f(x) + \langle g,h \rangle + o(h)$. Since $f$ is real valued, and $h \in \mathbb{C}$ the only way that $f(x+h)$ can be real valued for all $h$ is if $g \neq 0$.
@Paddy The question is what you mean by gradient. With functions of $z$ there are both $z$ and $\bar z$ derivatives unless the function is holomorphic or anti-holomorphic.
oh, just to check my brain on vector space stuff. Suppose I have two vector spaces with known bases. If I tensor those spaces together, do i get a basis by just tensoring all basis vectors in pairs?
@Paddy for a comparison, suppose you want to minimize $\|x+iy\|$ subject to $x\leq 3$. In terms of real variables, that amounts to minimizing $\sqrt{x^2+y^2}$ with $x\leq 3$ as a constraint.
so the relevant gradient in that case would just be the usual two-variable gradient in $x,y$
you can represent that in terms of complex numbers, but ultimately it's just doing stuff with real variables
@TedShifrin: Neighborhoods of isolated singular points (or more generally, normal spheres of higher-dimensional singularities) in projective varieties are naturally "contact manifolds". Why do I never see this in literature?
My days of working on CR geometry are long gone. Let's see — I guess integrability of that distribution isn't necessarily the contact condition. The canonical example is $\Re z_n = 0$ in $\Bbb C^n$.
the joke when Michel was presenting this stuff early on: "Yes, MJ the Younger exists and is not just a persona invented by MJ the Elder"
(basically, we focus on correlations that can be generated by the singlet state, which makes life easy because we don't have to worry about marginal probabilities and everything boils down to "hurr durr positive operators form a convex set". the moment you try to generalize, you have to worry about the marginals and there's no nice way to do that.)
i mean, that's definitely the source the Chinese would prefer
(and given that the main news articles on said theory are the Global Times, which wikipedia describes as "a daily tabloid newspaper under the auspices of the Chinese Communist Party's flagship People's Daily newspaper"...i'm not touching it until more objective sources pick it up)
from there you can define the expectation values of these operators as $E[A\otimes B]=\operatorname{tr}[\rho(A\otimes B)]$
and there's a total of eight such expectations to compute, for those four operators and the products $A\otimes B, A\otimes B',A'\otimes B,A'\otimes B'$
the challenge is to find a way to characterize those eight expectations, for generic $\rho$, knowing only information about the four operators
and boy is it tuogh
the one nice thing is that you can think of these operators as acting on the tensor product of pairs of two-by-two matrices
in which case there's a nice inner product structure via $\tr[AB]$, and writing out the basis 'vectors' for this hilbert space is easy
