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Mar
17
comment Why does $f = u+iv$ holomorphic $\implies$ $-if = -iu + v$ holomorphic?
this is a greedy answer. I like it, but, perhaps he ought to think about the CR-equations for the original function in comparison to the new multiplied function.
Mar
13
comment Special linear lie algebra relative with Angular Momentum
I seem to recall $L_1,L_2,L_3$ involve complex entries (I guess there are a few different representation choices here, but, I think they all have complex entries in at least one of these, maybe two) so what are you saying about $g$? Is it that you want to show $g$ is not Lie algebra isomorphic to $sl(2, \mathbb{R})$ ? I don't think the real span of the angular momentum is closed under bracket (need complex scalars)...
Mar
13
comment How to prove that $\{a\} \times \{a\} = \{\{\{a\}\}\}$
$\{ a \} \times \{ a \}$ not same as $a \times a$.
Mar
9
comment Determining the rank of a sparse tensor in $\mathbb{R}^{2 \times 2 \times 2}$
Thanks, I haven't really answered your question yet, but perhaps I made some progress towards a method. I need to think about how to formulate the rank-one-basis condition on an arbitrary subspace in the 8-dimensional space of third order tensors.
Mar
9
answered Determining the rank of a sparse tensor in $\mathbb{R}^{2 \times 2 \times 2}$
Mar
9
comment Determining the rank of a sparse tensor in $\mathbb{R}^{2 \times 2 \times 2}$
Thanks, although, I'm about to go to lunch here, so I may lose my chance to respond pretty soon...
Mar
9
comment Determining the rank of a sparse tensor in $\mathbb{R}^{2 \times 2 \times 2}$
I'm interested in your question, I'm not sure what the notation you use to define $T$ means. Could you elaborate on the initial notation used to define $T$?
Mar
5
comment Show that the linear function f(x)=Ax is differentiable at a
To begin, the fundamental theorem of linear algebra tells you that there exists $A$ an $n \times n$ matrix for which $f(x)=Ax$. Of course, $A$ is just the standard matrix of $f$ as we know and love from linear algebra...
Mar
5
answered Show that the linear function f(x)=Ax is differentiable at a
Mar
1
comment Expand $(\vec{A}\times \nabla)\times \vec{B}$ using tensorial notation
I'm sorry, I have trouble with this one. I need to stew on it a while, perhaps someone will swoop in and solve it before I finish this.... I haven't found the right way to write the operator expression in the Einstein notation just yet.
Mar
1
comment Expand $(\vec{A}\times \nabla)\times \vec{B}$ using tensorial notation
I think we need to antisymmetrize on the $\vec{A} \times \nabla$ term as to pick up the product rule. Unfortunately the usual formula $\vec{A} \times \vec{D}_k = \epsilon_{ijk}A_iD_j$ assumes that $\vec{A}$ and $\vec{D}$ have commuting components. Of course in this context that is false as $\vec{D}= \nabla$ is an operator so... we have to respect that.
Mar
1
comment Expand $(\vec{A}\times \nabla)\times \vec{B}$ using tensorial notation
I'm not yet sure how to do it with the tensorial approach, I need to ponder it a bit, but, I hope you can see from my current half-answer why there are three terms. You get one from raw derivatives acting on $\vec{B}$ then the terms like $\partial_2A_3$ act on $\vec{B}$ under a product rule hence producing two terms; $2+1=3$.
Mar
1
answered Expand $(\vec{A}\times \nabla)\times \vec{B}$ using tensorial notation
Feb
27
comment Multi-complex arithmetic in MATLAB?
I should talk to you more about what you are doing... let me digest that paper a bit.
Feb
27
comment Multi-complex arithmetic in MATLAB?
I don't know about any built-in commands, but, I could derive a matrix representation that you could build in matlab if interested.
Feb
26
comment Goursat's Theorem and Real Differentiability
you should read math.stackexchange.com/questions/833471/…
Feb
26
comment Goursat's Theorem and Real Differentiability
as the comment above might lead you... note that the difference between real and complex differentiability in the plane is the fact that the complex case allows the differential to be written as a multiplication of a complex number. In contrast, real differentiable allows many other possibilities, you need two complex numbers and the product of $w-z$ as well as the conjugate to capture real differentiability... sorry if this comment is cryptic, I'll try to find something to link.
Feb
25
comment Is it ever easier to show differentiability than continuity?
for example: math.stackexchange.com/q/447104/36530
Feb
25
answered Is it ever easier to show differentiability than continuity?
Feb
23
comment Reduction of order and lost in arithmetic
I haven't worked out this problem recently, but, perhaps multiplying by $\sqrt{x}$ then differentiating would make the calculation of $y'$ and $y''$ less annoying. That said, you are aware there is a formula for generating the second solution from the first for such a problem. However,the formula is not easy either. See page 86 of supermath.info/DifferentialEqns.pdf