It's more of a question; how do I enter an inline math? I can indeed enter \[\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 S}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0, \] but I cannot (or don't know how) to use a lousy $\varepsilon$ inline!

like this [$]\varepsilon[$] perhaps?

you put a dollar sign inside square brackets either side of the expression

[ $ ] \varepsilon [ $ ]

but with no spaces (and note there is no \ )

you put a dollar sign inside square brackets either side of the expression

[ $ ] \varepsilon [ $ ]

but with no spaces (and note there is no \ )

- Cuchulainn
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[$]\frac{\partial^3 u}{\partial x \partial y \partial z} = f[$]

Anyone know if this PDE has a name?

Anyone know if this PDE has a name?

I've never come across [$]\frac{\partial^3 u}{\partial x \partial y \partial z} = f[$], which doesn't mean that it doesn't have a name

if it helps (it probably doesn't) the 2D version is a version of the poisson equation

[$]\nabla^{2}u=f[$] becomes [$]4\frac{\partial^2 u}{\partial z\partial\bar{z}} = f[$] with [$]z=x+iy[$] and [$]\bar{z}=x-iy[$]

if it helps (it probably doesn't) the 2D version is a version of the poisson equation

[$]\nabla^{2}u=f[$] becomes [$]4\frac{\partial^2 u}{\partial z\partial\bar{z}} = f[$] with [$]z=x+iy[$] and [$]\bar{z}=x-iy[$]

- katastrofa
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That's something I would expect to find in the description of a fractal growth patterns in 3D, fluid mixing, amorphous or organic body surface formation, etc.

- Cuchulainn
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In all cases it is some kind of hyperbolic PDE. In the case of two independent variables we can transform

[$]\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}[$]

to

[$]\frac{\partial^2 u}{\partial \xi \partial \eta } [$]

by the change of coordinates

[$]x = (\xi + \eta), y = (\xi - \eta)[$]

Now the open question is if this works in 3 independent variables. Not sure if an uneven number of independent variables carries over.

[$]\frac{\partial^2 u}{\partial x^2} [$] [$] - \frac{\partial^2 u}{\partial y^2}[$] [$] - \frac{\partial^2 u}{\partial z^2}[$]

This PDE looks more benign for some reason

[$]\frac{\partial^2 u}{\partial x^2} [$] [$] - \frac{\partial^2 u}{\partial y^2} - \frac{\partial^2 u}{\partial z^2} - \frac{\partial^2 u}{\partial p^2}[$]

I think applications can be found in anisotropic waves.

// BTW the second derivatives are a bit shifted. It that the way it works here or am I missing a bracket etc?

[$]\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}[$]

to

[$]\frac{\partial^2 u}{\partial \xi \partial \eta } [$]

by the change of coordinates

[$]x = (\xi + \eta), y = (\xi - \eta)[$]

Now the open question is if this works in 3 independent variables. Not sure if an uneven number of independent variables carries over.

[$]\frac{\partial^2 u}{\partial x^2} [$] [$] - \frac{\partial^2 u}{\partial y^2}[$] [$] - \frac{\partial^2 u}{\partial z^2}[$]

This PDE looks more benign for some reason

[$]\frac{\partial^2 u}{\partial x^2} [$] [$] - \frac{\partial^2 u}{\partial y^2} - \frac{\partial^2 u}{\partial z^2} - \frac{\partial^2 u}{\partial p^2}[$]

I think applications can be found in anisotropic waves.

// BTW the second derivatives are a bit shifted. It that the way it works here or am I missing a bracket etc?

where I've come across the 2D version is Liouville's equation

- Cuchulainn
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ppauper wrote:where I've come across the 2D version is Liouville's equation

This is similar; in this case it is a concatenation of 1st order operators

[$]\frac{\partial }{\partial z} - i\frac{\partial }{\partial \overline{z}}[$] and [$]\frac{\partial u}{\partial z} + i\frac{\partial u}{\partial \overline{z}}[$]. This is is the elliptic case.

The hyperbolic case is similar but in real space.

[$]\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}[$] = [$](\frac{\partial }{\partial x} - \frac{\partial }{\partial {y}})[$] [$](\frac{\partial u}{\partial x} + \frac{\partial u}{\partial {y}})[$].

Then [$]x = (\xi + \eta), y = (\xi - \eta)[$] leads to [$]\frac{\partial^2 u}{\partial \xi \partial \eta } [$] .

Putting in a convection/drift term seems to cause meltdown fubar.

//

used \overline for complex conjugation, better than \bar?

- Cuchulainn
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test

[$]\displaystyle\lim_{n\to\infty} (1 + r/n)^{nt}[$]

[$]\displaystyle\lim_{n\to\infty} (1 + r/n)^{nt}[$]

- Cuchulainn
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A good friend is trying to create _general_ bidiagobnal matrix in LATEX

https://en.wikipedia.org/wiki/Bidiagonal_matrix

(with difficulty). Any templates for this?

Tridiagonal matrices are OK

https://en.wikipedia.org/wiki/Bidiagonal_matrix

(with difficulty). Any templates for this?

Tridiagonal matrices are OK

what exactly do you want, a lot of the entries replaced by [$]\cdots[$] or [$]\vdots[$] or [$]\ddots[$]?

- Cuchulainn
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ppauper wrote:what exactly do you want, a lot of the entries replaced by [$]\cdots[$] or [$]\vdots[$] or [$]\ddots[$]?

The prototypical example is the upper bidiagonal matrix here.

https://www.quantstart.com/articles/Tri ... -Algorithm

Cuchulainn wrote:ppauper wrote:what exactly do you want, a lot of the entries replaced by [$]\cdots[$] or [$]\vdots[$] or [$]\ddots[$]?

The prototypical example is the upper bidiagonal matrix here.

https://www.quantstart.com/articles/Tri ... -Algorithm

in firefox, when you right click on a picture, you can copy the URL

when I right click on the equation on that site, it gives me the tex

[$]\begin{bmatrix}

1 & c^{*}_1 & 0 & 0 & ... & 0 \\

0 & 1 & c^{*}_2 & 0 & ... & 0 \\

0 & 0 & 1 & c^{*}_3 & 0 & 0 \\

. & . & & & & . \\

. & . & & & & . \\

. & . & & & & c^{*}_{k-1} \\

0 & 0 & 0 & 0 & 0 & 1 \\

\end{bmatrix} \begin{bmatrix}

f_1 \\

f_2 \\

f_3 \\

.\\

.\\

.\\

f_k \\

\end{bmatrix} = \begin{bmatrix}

d^{*}_1 \\

d^{*}_2 \\

d^{*}_3 \\

.\\

.\\

.\\

d^{*}_k \\

\end{bmatrix}

[$]

- Cuchulainn
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That works great. Thanks.

We added some icing by printing the ellipsis .. . along the diagonals.

We added some icing by printing the ellipsis .. . along the diagonals.