JOST A MON: typesetting

Nov 11, 2012

A Theorem

It's been a while since I've CodeCogged, and I'm pleased to find that the site is still there. This brings out an intemperate desire to see some fancy mathematical typesetting, and an email from Springer provides the impetus. I get free previews for the occasional paper, you see, and it's always fun to look at its contents and realise, once again, that while I might understand every word in a sentence, I have no idea what the thing in its entirety means.

So, anyway, here's a theorem from Alexander Kuznetsov (2008) 'Lefschetz decompositions and categorical resolutions of singularities', Selecta Mathematica (New Series), Volume 13, Number 4, pp 661-696.

Theorem 1. The triangulated category $\widetilde{D}$ with functors $\pi_{*}$ and $\pi^{*}$ is a categorical resolution of $D^{b}\left ( Y \right )$ . Moreover, we have a semiorthogonal decomposition

$D^{b}\left ( \widetilde{Y} \right )=\left \langle i_{\bullet} \left \left ( B_{m-1} \otimes L^{1-m} \right )\right,\left ( B_{m-2} \otimes L^{2-m} \right ), \cdots ,\left ( B_{1} \otimes L^{-1} \right ),\widetilde{D}\right \rangle.$

Finally, if $Y$ is Gorenstein and $K_{\widetilde{Y}}=\pi^{*}K_{Y}+\left ( m-1 \right )\widetilde{Z}$ then

$\widetilde{D}$ is weakly crepant if $p^{*}\left ( D^{perf}\left ( Z \right ) \right )\subset B_{m-1},$

$\widetilde{D}$ is strongly crepant if $B_{m-1}= B_{m-2}=\cdots=B_{1}=B_{0}$

Kuznetsov goes on to say: 'Certainly, minimal categorical resolutions are the most interesting.'

One day I'll figure out how to inline the LaTeX so that it doesn't appear off the rest of the text. Good thing to have an occasional challenge, eh?

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Mar 26, 2008

Code Cogs

So! It appears that there is one (still clumsy) way to display math text in Blogger. The folks at CodeCogs have to be thanked for this.

$\int_{-\infty }^{\infty }e^{-x^{2}}\;dx=\sqrt{\pi }$

It's still rather painful in that if I switch to the WYSIWIG editor, I get an INVALID EQUATION warning, and when I come back to the HTML editor, I find that the source line invoking Code Cogs' parser and image creator is garbled. Sheesh.

But it doesn't look half-bad, eh? A bit pixellated, but clear enough.

$\be P_R = \frac{\partial L}{\partial\dot R} = - \frac{6R \dot R}{\kappa^2 N}, \qquad P_{\phi} = \frac{R^3 \dot \phi}{N}. \ee$

Or even:

$\be S_{(n=2)} &=& S_g + S_m = \int \left( -\frac{3}{\kappa^2}{I\!\!N}^{-1} {I\!\!R} D_{\bar\theta} {I\!\!R} D_{\theta}{I\!\!R} + \frac{3 \sqrt k}{\kappa^2} {I\!\!R}^2 \right) d\theta d\bar\theta dt \ee$

Royal pain, though, to have to put the Latex code into one long line without breaks.

And an even bigger problem is, of course, that each time this page is rendered, a query goes out to the CodeCogs server, causing it to shudder. This is not a nice thing to do, so I'll retire this page tonight. (Ah, but see Update below.)

So! The search for built-in math typesetting in Blogger continues...

UPDATE 26/03/2008: Got permission from the folks at CodeCogs to add the aforementioned links. Currently they can soak up the load, but if the usage increases dramatically, they expect to ask for donations. Hence, this page goes back up.

Also, it appears that the problem of garbling the TeX code when switching from HTML to WYSIWIG is a bug in the Internet Explorer: it doesn't like the TeX code embedded within HTML if there are spaces in the TeX, and truncates at the first space. Firefox doesn't have this problem. CodeCogs has a tool that generates TeX, and will post a fix such that the TeX it generates will not have spaces. That's not a full fix for the problem of course - Internet Explorer needs to sort out its HTML parsing, I guess. Meanwhile, use Firefox and remember to brush your teeth after every meal!

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Jan 6, 2008

Posting From Live Writer

Looks like the beavers at Microsoft have been a bit busy. They have created this free tool called Live Writer that enables the user to post to a blog without having to login to the blog's website. Supposedly it provides better format control than, say, Blogger, but I still don't see the ability to typeset mathematics! So my frustration, which started here, continues.

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Aug 21, 2007

Numerically Typing

Came across a nifty piece on shifting standards in numerical typesetting.

If your numerals are embedded in text, use what the printers call old-style fonts: variable-width fonts with ascenders and descenders. For example, Georgia: 0123456789.

For standalone numbers, use modern fonts: fixed-width and equally sized. For example, Courier: 0123456789.

Helvetica is abhorred by business card makers because the constant-sized numerals make it difficult to differentiate among them. Check this out: 0123456789.

Interested in a brief history of typesetting? Here is an accessible one.

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Apr 14, 2007

Math in a Blog

If there are physicists blogging away with the best of them, surely mathematicians can't be far behind? How do they all typeset all the formulae and algebra and equations? I'm amazed that a cutting-edge outfit such as Google still hasn't managed to incorporate a mathematical typesetter into Blogger.

A quick search (on Google, where else?) reveals that the Wordpress.com blogging site now supports an inline L^AT_EX command parser. L^AT_EX, as you all know, is the de-facto standard for mathematical publishing. I haven't used it since my M.Engg. thesis back in 1994, and am ashamed to say that I've chickened out of it recently, and have been formulising with Microsoft's Equation Editor.

A naive and ugly way is to write the mathematics in L^AT_EX, convert it to an intermediate format (say, GIF?) file and import it into the blog.

Ostensibly, Google Docs can export documents to Blogger, but the former doesn't provide native math typesetting, so this project may be of some use. In my previous post on the Tengwar, I mentioned my frustration with the incorporation of that font into Blogger; this lack of mathematical typesetting is equally frustrating.

An alternative methodology seems to be the use of MathML. Firefox supports this natively. Wa-hey! So what now? This bears thinking about.

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Apr 7, 2007

The Tengwar of Fëanor

Right. So Tolkien, in his infinite wisdom and fecund imagination, concocted languages of superlative beauty and expressiveness. Not content with mere grammar and linguistics, he designed fonts as well. Or rather, he reported that Fëanor created them when he could spare time from his life's work - the Silmarils, wrought from the light of the Two Trees, blessed by Yavanna, coveted and stolen by Morgoth, and the cause of the long war and bitter defeat of the High Elves in Middle Earth.

Heady stuff. And considering that the appendices to the Lord of the Rings, and the Silmarillion, and the collected notes of Tolkien compiled by his son contain so much information on the languages of his worlds, one would think that there would be little more to add by others. But one would think wrong:

A Dictionary of Sindarin
All the languages of Tolkien.
Fonts, galore.

I've been trying with a singular lack of success to post something in the Tengwar font. I even thought I could upload a document with the appropriate font into Google Docs. But no. All I get is: Words -- so innocent and powerless as they are, as standing in a dictionary, how potent for good and evil they become in the hands of one who knows how to combine them. [N. Hawthorne]

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JOST A MON

The idle ramblings of a Jack of some trades, Master of none

A Theorem

Code Cogs

Posting From Live Writer

Numerically Typing

Math in a Blog

The Tengwar of Fëanor

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